algebraic geometry notes - columbia universitystanislav/notes/algebraic... · 2018. 11. 25. ·...

ALGEBRAIC GEOMETRY NOTES

STANISLAV ATANASOV

The following notes came to life from my attempts to learn algebraic geometry during a year-longsequence of classes taught by Prof Johan de Jong at Columbia I would be very happy if theyhelp even one other person in his or hers first steps in the tantalizing world of algebraic geometryMany of the results are stated in the generality presented in class Some additional topics such as(faithful) flatness and group law on elliptic curves were added later The primary sources are [1]and [6] One can find most of the proofs there although not always in the same generality as theone presented here Occasionally the exposition may follow closely some other sources - these arelisted in the bibliography These notes were written with the intention to serve as a repository ofresults and not as a self-contained lecture series as a result barely any proofs are presented Theorganization is representative of my taste and intuition and I apologize if it makes little sense toothers To partially offset that I have included a detailed list of contents

None of the results but all of the mistakes are mine If you find any mistakes - both typographicalor serious mathematical ones - please email me at stanislavmathcolumbiaedu

1

2 STANISLAV ATANASOV

Contents

1 Definitions and formulae 411 Most common definitions 412 Operations on sheaves 613 Dimension theory 714 Spectral sequences 715 Euler characteristic and polynomial 816 Computations of cohomology 917 Divisors 1118 Formulas 1119 Intersection theory after blow-up 122 Fiber product of schemes 1421 Properness and separatedness 1522 Flatness and faithful flatness 153 Numerical invariants 1831 Modules of differentials 1832 Hurwitzrsquos theorem 2233 Serre duality Normal and conormal sheaves 224 Tangent spaces 2541 Tangent space as derivations 2542 Tangent space via dual numbers 255 Line bundles and divisor correspondence 276 Computations of grd 3061 Separating points and tangent vectors 3062 Commonly used facts 327 Curves 3371 Maps between curves 3472 Curves and divisors 3473 Genus 0 3674 Genus 1 3675 Genus 2 3776 Genus 3 3777 Genus 4 3778 Other results for curves 388 Surfaces 3981 First approach to intersection theory 3982 Numerical intersections 4083 Divisors 4184 Ampleness 449 Blow-ups 4691 Universal properties and intersection theory 4892 Comparison with cohomology of the blow-up 4910 Classification of surfaces 5011 Riemann hypothesis for curves over finite fields 5212 Abelian varieties 55121 Basic properties of abelian varieties 55122 Complex tori 56

ALGEBRAIC GEOMETRY NOTES 3

123 Theorem of the cube 57124 Picard scheme dual abelian variety and polarizations 59125 Tate modules and `minusadic representation 63126 Poincare complete reducibility and Riemann-Roch for abelian varieties 65127 Weil pairing and Rosati involution Classification of End(A) 66128 Abelian varieties over finite fields 69129 Complex multiplication 73References 75


1 Definitions and formulae

11 Most common definitions

Definition 111 A scheme X is

(1) reduced if OX(U) has no nilpotent elements forallU -open(2) integral if OX(U) is an integral domain forallU -open

integral = reduced + irreducible

(3) noetherian if X = cupNi=1 Spec(Ai) with Ai-Noetherian(4) quasi-compact if every open cover has a finite subcover(5) quasi-separated if intersection of two affine open sets is finite union of affine open sets(6) separated if the intersection of two quasi-compact open sets is quasi-compact We also

have intersection of two affine open sets is affine

X is qcqs hArr X can be covered by finitely many affine opens any two of whichhave intersection also covered by finitely many affine open subsets

(7) quasi-affine if it is quasi-compact and isomorphic to an open affine scheme ie X Spec(A)V (I)where A is a ring and I = (f1 fr) is a finitely generated ideal

(8) projective over Y if Y is affine and there exists n ge 0 and a closed immersion X minusrarr PnY1

over Y (9) normal if the local ring OXx is an integrally closed domain for any point x isin X

(10) a variety if it is a separated reduced irreducible scheme of finite type over algebraicallyclosed field k

(11) a curve if it is a regular integral scheme of dimension 1 proper over a field k(12) a surface if it is a variety of dimension 2

For schemes XY let ∆XY X rarr X timesY X be the morphism such that pri ∆XY = idX

Definition 112 A morphism of schemes f X rarr Y is

(1) closed immersion if f is a homeomorphism onto a closed subset of Y and f OX rarrflowastOY is surjective

(2) separated if ∆ X rarr X timesY X is closed immersion(3) closed if the image of a closed set is closed(4) universally closed if it is closed and for every morphism of algebraic spaces Y prime rarr Y the

corresponding X prime rarr Y prime after base change is also closed

X prime

X

f

Y prime Y

(5) proper if it is separated of finite type and universally closed(6) projective if it factors through a closed immersion i X rarr PnY for some n isin N and affine

Y (7) affine if the inverse image of every affine open of Y is an affine open of X Also if f is

qc+s then f is affine iff flowast is exact in QCoh

An affine morphism is separated and quasi-compact

1if Y = SpecB then PnY = Proj(B[T0 Tn]

)


(8) locally of finite type if there exists affine covering Y = cupVi Vi = Spec(Bi) st fminus1Vi =cupUij with Uij = SpecAij and Aij is finitely generated over Bi

(9) finite type if locally of finite type and each fminus1(Vi) has a finite cover Uij(10) finite if it is affine and forall U sube Y open the ring homomorphism OY (U)rarr OX(fminus1(V )) is

finite ie OX(fminus1(V )) is a finite OY (V )-module(11) smooth at x isin X if there exists an affine open neighbourhood Spec(A) = U sub X of x and

affine open Spec(R) = V sub Y with f(U) sub V such that the induced ring map R rarr A issmooth

(12) smooth if it is smooth at every point of X(13) flat at x isin X (resp faithfully flat at x isin X) if the induced map flowast OYf(x) rarr OXx is

flat (resp flat and f is surjective)2

(14) flat (resp faithfully flat) if it is flat (resp faithfully flat) at every point of X

Remark 111

bull An S-scheme X has property P lArrrArr the morphism X rarr SpecS has property P bull A k-scheme X has rdquogeometrically property Prdquo lArrrArr Xkprime has proeprty P for every field

extension k of k

Lemma 113 (Affine communication lemma [7 Lemma 532]) Let P be some property enjoyedby some affine open subsets of a scheme X such that

i) if P holds for U = SpecA sub X then for any f isin A property P holds for D(f)ii) If U = U1cupmiddot middot middotUn where U = SpecA and Ui = D(fi) and P holds for Ui then P holds for U

Then P holds for every affine of X

This lemma can be applied in many situations For instance

Example 111 (Affine local on the target) The morphism f X rarr Y is

(1) quasi-compact if fminus1(Ui) is quasi-compact on some affine cover Ui of Y (2) quasi-separated if fminus1(Ui) is quasi-separated on some affine cover Ui of Y (3) separated if fminus1(Ui) is separated on some affine cover Ui of Y (4) finite if fminus1(SpecAi) = SpecBi with Bi finite Ai-algebra on some affine cover SpecAi

of Y (5) locally of finite type if fminus1(SpecAi) is locally finite type over Bi on some affine coverSpecAi of Y

Lemma 114 (Induction principle) Let X be a quasi-compact and quasi-separated scheme LetP be a property of the quasi-compact opens of X Assume that

i) P holds for every affine open of Xii) if U is quasi-compact open V affine open P holds for U V and U cap V then P holds for

U cup V

Then P holds for every quasi-compact open of X and in particular for X

2a well-known result states that for local rings flatness is equivalent to faithful flatness ie ϕ Ararr B is flat andϕlowast Spec(B)rarr Spec(A) is surjective This implies that OYf(x) rarr OXx is injective


12 Operations on sheavesLet f X rarr Y be a continuous map of topological spaces

Definition 121 The pushforward or direct image sheaf is given by

flowast ShX rarr ShY

F(U) 7rarr flowastF(U) = F(fminus1(U)

)Given a morphism of sheaves α F rarr G we have

flowast(α) flowast(F)rarr flowast(G)

F(fminus1(V )

)3 s 7rarr αfminus1(V )(s)

Example 121

(1) If f X rarr lowast then flowastF = F(X) = Γ(XF) the global sections(2) Let x isin X Consider i x minusrarr X Then ilowastA is the skyscaper sheaf at the point x

Remark 121 The functor flowast is left exact It is exact if f = i X minusrarr Y is inclusion of closedsubspace

Definition 122 The pullback or inverse image sheaf is the sheafification of the presheaf

fminus1 ShY rarr ShX

U 7rarr colimVsupef(U)

F(V )

Example 122

(1) If i x minusrarr X then iminus1F = Fx the stalk at x(2) If p X rarr lowast then pminus1(Zlowast) = ZX (3)

(fminus1F

)x

= Ff(x)

Remark 122 The functor fminus1 is exact No sheafification required if f is inclusion of open set

Definition 123 Let f X rarr Y be a morphism of locally ringed spaces For a OY -module G theinverse image functor is given by

flowastG = fminus1Gotimesfminus1OY OX

Remark 123 The functor flowast is right exact When exact the map f is called flat

Definition 124 Let j U minusrarr X be inclusion of open subset For F isin ShU we construct theextension by zero sheaf

jF(V ) =

F(V ) if V sub U0 otherwise

Remark 124 The functor j is exact The sheaf jF is the unique sheaf FU on X such that

FU∣∣U

= F and FU∣∣XminusU = 0


Definition 125 The support of a sheaf F isin ShX is given by

suppF = x isin X Fx 6= 0For a section s isin F(U) the support of this section is

supp s = x isin X sx 6= 0

Remark 125 Support of a section is always closed subset of U while support of a sheaf may notbe closed The support of coherent sheaf on a Noetherian scheme is always closed3

13 Dimension theory

Lemma 131 (Trdeg is additive) If LFk is a tower of field extensions then trdegk(L) =trdegk(F ) + trdegF (L)

Lemma 132 Let k be a field and f isin k[x1 xn] be an irreducible polynomial If K =Frac

(k[x1 xn](f)

) then trdegk(K) = nminus 1

Lemma 133 (Prime avoidance) Let R be a ring and I C R be an ideal Let pi C R be primeideals If I sube cupipi then I sube pk for some k

In an affine scheme if a finite number of points are contained in an open subsetthen they are contained in a smaller principal open subset

Definition 134 Let X be a topological space A function δ X rarr Z is called dimensionfunction if

(1) x y4 and x 6= y then δ(x) gt δ(y)(2) x y is an immediate specialization then δ(x) = δ(y) + 1

Theorem 135 (Noether normalization) Let k be a field Suppose 0 6= A is a finite type k-algebraThen there exists n ge 0 and a finite injective k-algebra map

k[x1 xn] minusrarr A

Theorem 136 (Dimension function) Let k be a field and A be a finite type algebra over k Then

δ Spec(A)rarr Zp 7rarr trdegk(κ(p))

is a dimension function Moreover δ(p) = 0hArr p is a maximal ideal hArr κ(p)k is a finite extension

Theorem 137 (Hilbert Nullstellensatz) Each point is given by the polynomials that vanish atthat point ie

p = capxisinXmx

14 Spectral sequences

Theorem 141 (Leray spectral sequence) Let f (XOX) rarr (YOY ) be a morphism of ringedspaces For any OX-module F there is a spectral sequence

Hq(YRpflowastF)rArr Hp+q(XF)

3if M is finitely generated over Noetherian R then supp(M) = V (AnnR(M))4ie y isin x


Theorem 142 (Local-global spectral sequence for Ext) Suppose X is a ringed space and FGare OX-modules There exists a spectral sequence

Hp(XExtq(FG)

)rArr Extp+q

(FG

)

Corollary 141 If Ext iOX (FG) = 0 for i = 0 l minus 1 then

ExtlX(FG) = H0(XExt lOX (FG)) = Γ(XExt lOX (FG)

)

Corollary 142 If E is finite locally free then Ext iOX (EG) = 0 i gt 0 Hence

ExtiX(EG) = H i(XHomX(EG)) = H i(XEor otimesOX G)

15 Euler characteristic and polynomial

Let k be a field and let X is a projective scheme over k ie Ximinusrarr Pnk is a closed subscheme Set

OX(d) = ilowastOPnk (d)

Definition 151 For coherent F on X we set the Euler characteristic of F on X to equal

χ(XF) =infinsumi=0

(minus1)i dimkHi(XF)

Proposition 151 (Hilbert polynomial of F) For F isin Coh(X) the function PX given by

d 7rarr χ(XF(d)

)is numerical polynomial (ie for large d it becomes a polynomial) of degree at most dim

(suppF

)5

Theorem 152 (Serre vanishing) For any coherent F on a projective scheme X H i(XF(m)

)= 0

for all i gt 0 whenever m 0

Remark 151 Theorem 152 implies that the Euler characteristic eventually coincides with bot-tom cohomology (global sections) for projective schemes

Example 151 The Hilbert polynomial POPnk

of OPnk is(n+tt

)isin Q[t]

Example 152 If F isin Coh(X) has PF equiv 0 then F equiv 0

Example 153 Let PF(t) be the Hilbert polynomial of F isin Coh(X) Then

(1) PF(d)(t) = PF(t+ d)(2) PFoplusG(t) = PF(t) + PG(t)(3) (Euler characteristic is additive) If 0 rarr K rarr F rarr G rarr 0 is exact sequence of OX-

modules then PF(t) = PK(t) + PG(t)(4) If X sub Y sub Pnk are closed embeddings then PX(t) le PY (t) If PX(t) = PY (t) for t 0

then X = Y 6

Example 154 Suppose H is a degree d hypersurface in Pn From the closed subscheme exactsequence

0 minusrarr OPn(minusd) minusrarr OPn minusrarr OH minusrarr 0

we derive the Hilbert polynomial

PH(t) = PPn(t)minus PPn(tminus d) =

(n+ t

n

)minus(n+ tminus d

n

)

5in this case the support is actually closed subset of X6ideal sheaf sequence for first part and Example 151 for second


Example 155 Let C be a degree d curve in P2k From Example 154 we obtain χ(COC) =

PC(0) = minusd(dminus3)2

7 We have

0 minusrarr OP2(minusd) minusrarr OP2 minusrarr OC minusrarr 0

Since H1(P2OP2(minusd)) = 0 the sequence is exact when passing to global sections Hence as dim isadditive on exact sequences we obtain

dimH0(COC) = dimH0(P2OP2

)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1

Lastly χ(COC) = dimH0(COC)minus dimH1(COC) so

dimH1(COC) =(dminus 1)(dminus 2)

2

16 Computations of cohomology

Theorem 161 (Grothendieck) Let X be a Noetherian topological space If dim(X) le d thenHp(XF) = 0 for all p gt d and any abelian sheaf F on X

Lemma 162 If a scheme X can be covered by n affine open sets then Hi(XF) = 0 for i ge nand all F

Theorem 163 (Cohomology of projective space) Let R be a ring Let n ge 0 be an integer Wehave

Hq(PnROPnR

(d)) =

(R[T0 Tn])d if q = 0

0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n

as R-modules In terms of dimension we have

bull dimH0(PnROPnR(d)

)=(n+dd

)if d ge 0

bull dimHn(PnROPnR(d)

)=( minusdminus1minusnminusdminus1

)if d le minusnminus 1


)= 0 otherwise

Definition 164 Given an open cover U U = cupni=1Ui and an abelian presheaf F consider thecomplex prod

i0

F(Ui0) minusrarrprodi0lti1

F(Ui0i2) minusrarr middot middot middot minusrarrprod

i0ltmiddotmiddotmiddotltinminus1

F(Ui0i2 inminus1) minusrarr F(Ui0in)

with differential map d(s)i0ip+1 =sump+1

j=0(minus1)jsi0ij ip+1

∣∣∣i0ip+1

The cohomology obtained from

this complex is called Cech cohomology

Lemma 165 Let X be a topological space U open cover of X and F sheaf of abelian groupsThen

H0(UF) Γ(XF)

This ceases to be true for higher cohomologies However we have partial salvation

Proposition 161 For any topological space X and any abelian sheaf F there is an isomorphism

limminusrarrU

H(UF)simminusrarr H1(XF)

where the colimit is taken over all refinements

7in particular for d gt 2 the curve C is not isomorphic to P1 as χ(P1OP1) = 1


Proposition 162 Let X be a topological space U = (Ui) open cover of X and F sheaf ofabelian groups Suppose that for any finite intersection V = Ui0in and for any k gt 0 we haveHk(VF

∣∣V

) = 0 then

Hp(UF)simminusrarr Hp(XF)

Theorem 166 [1 III Thm 45]) Let X be noetherian separated scheme U open affine cover ofX and F isin QCoh(X) Then for all p ge 0 we have a natural isomorphism8


Lemma 167 (Mayer-Vietoris) Let X be a ringed space Suppose that X = U cup V is a union oftwo open subsets For every OX-module F there exists a long exact cohomology sequence

0rarr H0(XF)rarr H0(UF)oplusH0(VF)rarr H0(U cap VF)rarr H1(XF)rarr

where

middot middot middot rarr H i(XF)rarr H i(UF)oplusH i(VF)rarr H i(U cap VF)rarr middot middot middotmiddot middot middot 7rarr s 7minusrarr s

∣∣Uoplus s∣∣V7minusrarr s

∣∣Uminus s∣∣V7rarr middot middot middot

This long exact sequence is functorial in F

Proposition 163 Let f X rarr Y be affine morphism with Y quasi-compact and separated (andhence so is X9) Then for any F isin QCoh(X) we have for each i ge 0

H i(Y flowastF) H i(XF)

Proposition 164 Let f X rarr Y be affine morphism with Xminus Noetherian Then for anyF isin QCoh(X) we have for each i ge 0


Remark 161 When f is a closed embedding (hence affine) and Y = PnA Proposition 163allows us to translate calculations from arbitrary A-scheme to PnA

Theorem 168 Let R be noetherian For any F isin Coh(PnR) we have

(1) H i(PnRF) is a finite R-module for all i(2) H i

(PnRF(d)

)= 0 for d 0 and i gt 0

(3) H i(PnRF) = 0 for i gt n(4) oplusdge0H

0(PnRF(d)

)is finite graded R[T0 Tn]-module

Theorem 169 Let A be Noetherian graded ring fin gen by A1 over A0 Then the statementof Theorem 168 holds for all coherent F on Proj(A)

Theorem 1610 (flowastQCoh = QCoh ) If f X rarr Y is qcqs then flowast preserves quasi-coherence

Lemma 1611 Cohomology on qcqs scheme X commutes with filtered colimits

H i(X limminusrarrFj) = limminusrarrH i(XFj)

for all i ge 0 In particular commutes with arbitrary direct sums

8follows by abstract nonsense but the map is functorial in F9affine morphisms are quasi-compact and separated


17 Divisors

Definition 171 Let the scheme X be regular in codimension one10 A prime divisor is a closedintegral subscheme Y sub X of codimension one A Weil divisor is an element of the free abeliangroup Div(X) generated by the prime divisors of X

Definition 172 Let i Y minusrarr X be a closed immersion The sheaf of ideals of Y is then thekernel of the surjection

IZ = ker(OX

iminusrarr ilowastOZ)

Remark 171 In this case there is a correspondence

sheaves on X annihilated by IY larrrarr sheaves on Y

Definition 173 A Cartier divisor on a scheme X is a global section of KtimesOtimes Equivalentlyit is given by a data of open cover Ui and

fi isin Γ(UiKtimes)

such that fifj isin Γ(Ui cap Uj Otimes) It is effective if it can be represented by Ui fi where fi isinΓ(UiO

∣∣Ui

) In this case we define associated subscheme of codimension 1 to be the closed

subscheme Y corresponding to the sheaf of ideals IY locally generated by fi

Remark 172 Cartier divisors on X thus correspond to locally principal closed subschemes

Proposition 171 Let D be an effective Cartier divisor on X with associated subscheme Y Then

OX(minusD) = IY

Definition 174 A linear system d is a subset of |D| corresponding to V sube Γ(XL) given by

V = s isin Γ(XL) (s)0 isin d cup 0

Definition 175 Let X be a projective curve A grd is a pair (L V ) where

bull L isin Picd(X)bull V sube H0(XL)- vector subspace of dimension r + 1

We say (L V ) is base point free if forallx isin Xexists isin V st s does not vanish at x Conversely wesay x is a base point of a linear system d if x isin suppD for all D isin d

18 Formulas

Theorem 181 (Riemann-Roch for Curves) For locally free sheaf E on X we have

χ(XE) = deg(E)minus rank(E) middot (g minus 1)

For E = L = OX(D) a line bundle

h0(OX(D))minus h1(OX(D)) = deg(OX(D)) + 1minus g

Combining with Serre duality

h0(OX(D))minus h0(ωX(minusD)) = deg(OX(D)) + 1minus g

10every local ring OXx of dimension one is regular


Theorem 182 (Riemann-Roch on surfaces) Let X be a smooth projective surface and L isinPic(X) Then

χ(XL) =1

2c1(L) middot c1(L)minus 1

2c1(L) middot c1(ωX) + χ(XOX)

In terms of divisor if D isinWeil(X) then

χ(XOX(D)) =1

2D middotD minus 1

2D middotKX + χ(XOX) =

1

2D middot (D minusK) + χ(XOX)

Remark 181 This means that the polynomial

n 7rarr χ(XLotimesn)

has leading term 12c1(L) middot c1(L) linear term minus1

2c1(L) middot c1(ωX) and constant term χ(XOX)

Theorem 183 (Serre duality for curves) For a smooth curve X over an algebraically closed fieldk with canonical sheaf ωX we have

h0(L) h1(ωX otimes Lminus1)


Proposition 181 (Serre duality for projective space) Let X = Pdk then

(1) Hd(XωX) k(2) for every i ge 0 there is a functorial isomorphism

Exti(F ω)simminusrarr Hdminusi(PdkF)or

Theorem 184 (Hurwitz) Let f X rarr Y be a finite separable morphism of curves with n = deg fThen

2g(X)minus 2 = n middot (2g(Y )minus 2) +sumpisinX

(ep minus 1)

Theorem 185 (Clifford) Let X be a smooth projective curve and L isin Pic(X) is such that h0(L)and h1(L) are both positive11 Then

h0(L) le degL

2+ 1

Furthermore if equality holds then one of the following is true

(1) L ωX (2) L OX (3) X is hyperelliptic curve L Notimesi for some 1 le i le g minus 2 where (N H0(N)) is g1

2

19 Intersection theory after blow-up

Proposition 191 (Intersection theory on X) Let π X rarr X be a blow-up at a closed point pThen

Pic X PicX oplus Zand

(1) the exceptional divisor E sub X is isomorphic to P1k

(2) if CD isin PicX then (πlowastC) middot (πlowastD) = C middotD(3) if C isin PicX then (πlowastC) middot E = 0

11such line bundles correspond to the so-called rdquospecial divisorsrdquo


(4) E2 = minus1

(5) if C isin PicX and D isin Pic X then (πlowastC) middotD = C middot (πlowastD) where πlowast Pic X rarr PicX is theprojection onto the first factor

(6) If C isin Pic(X) and degE(C∣∣E

) = 0 then πlowast(X) isin Pic(X) and C πlowastπlowastC(7) For n isin Z we have

πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


2 Fiber product of schemes

Lemma 201 If X rarr S larr Y are morphism of affine schemes corresponding to maps of ringsAlarr Rrarr B then X timesS Y is the affine scheme Spec(AotimesR B)

Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U

be a fiber product of schemes Let U sube S V sube X and W sube Y open such that p(V ) sube U supe q(W )Then

V timesU W = pminus1(V ) cap qminus1(W ) sube X timesS Y

Lemma 203 Let

X timesS Uq

p

U

i

Xf

S

be a fiber product of schemes Suppose i U minusrarr S is an open embedding Then p X timesS U rarr X isalso open embedding Moreover

X timesS U sim=(fminus1(U)OX

∣∣fminus1(U)

)

Remark 201 One constructs X timesS Y as follows Choose affine open coverings S = cupUi andfminus1(Ui) = cupVj and gminus1(Ui) = cupWk Then

X timesS Y =⋃Vj timesUi Wk

glued along open intersections as described in Lemma 202

Example 201 We have PnR = PnZ timesSpecZ SpecR and AnR = AnZ timesSpecZ SpecR

Proposition 201 ([7 Ex 94 B]) The following properties of morphisms are preserved underbase-change

i) closed embeddingii) open embedding

iii) quasi-compactiv) quasi-separatedv) affine

vi) finitevii) integral

viii) locally of finite type

Remark 202 In light of Remark 201 we can compute fibre products in four steps

(1) Base change by open embedding This is the content of Lemma 203(2) Adding extra variable Recall that AotimesB B[x] sim= A[x](3) Base change by closed embedding For a ring morphism ϕ B rarr A and ideal I B

we have AIe sim= AotimesB (BI) where Ie = 〈ϕ(i)〉iisinI Also Ie = I otimesB A(4) Localization For a ring morphism ϕ B rarr A and a monoid S sube B we have ϕ(S)minus1A sim=

AotimesB Sminus1B


21 Properness and separatedness

Definition 211 Let S be a fixed base scheme If Sprime is another base scheme and Sprime rarr S ismorphism then X timesS Sprime is base extension

X timesS Sprime

B

Sprime S

Lemma 212 Any morphism f X rarr Y between affine schemes is separated

Corollary 211 A morphism f X rarr Y is separated if and only if ∆(X) sub X timesY X is closed

Theorem 213 (Properties of separated morphism) Assume all schemes are noetherian Then

bull Open and closed immersions are separatedbull Composition of separated morphisms is separatedbull Separated morphisms are stable under base extensionbull If f X rarr Y and f prime X prime rarr Y prime are two separated morphisms of schemes over S then

f times f prime X timesS X prime rarr Y timesS Y prime

is also separated

Theorem 214 (Properties of proper morphisms) Assume all schemes are noetherian Then

bull A closed immersion is properbull Composition of proper morphisms is properbull Proper morphisms are stable under base extensionbull (rdquo2-out-of-3 propertyrdquo) If f X rarr Y and g Y rarr Z are two morphisms and gf is proper

and g separated then f is properbull Projective morphism is necessarily proper

Almost all examples of proper morphisms will come from projective ones

Lemma 215 (Chowrsquos Lemma) Let S be a Noetherian scheme Let f X rarr S be a separatedmorphism of finite type Then there exists a diagram

X

X prime

πoo Pn

S

~~

S

where X prime rarr PnS is an immersion and π X prime rarr X is proper and surjective Moreover we may

arrange it such that there exists a dense open subscheme U sub X such that πminus1(U) rarr U is anisomorphism

22 Flatness and faithful flatness

221 For rings and modules

Definition 221 Let M be an A-module Then

bull M is flat if

0rarr N rarr Lminus exact =rArr 0rarr N otimesAM rarr LotimesAM minus exact


bull M is faithfully flat if

0rarr N rarr Lminus exactlArrrArr 0rarr N otimesAM rarr LotimesAM minus exact

Theorem 222 Flatness is a local condition That is M is flat if and only if Mm is flat for allmaximal ideals m

Theorem 223 Let A be a Dedekind domain Then an Aminusmodule M is flat if and only if it istorsion free

Theorem 224 Let A be a ring and M an Aminusmodule Then the following are equivalent

1 M is faitfully flat2 M is flat and N otimesAM for all 0 6= N3 M is flat and AmotimesAM 6= 0 for all maximal mCA

Corollary 221 Let ϕ (Am) rarr (B n) be a local ring map and let M be a finitely generatednonzero Bminusmodule Then M is faithfully flat if and only if it is flat

On the level of rings we have the following useful criterion

Theorem 225 Let ϕ Ararr B be a ring map Then the following are equivalent

1 ϕ is faithfull flat2 ϕ is flat and ϕlowast Spec(B)rarr Spec(A) is surjective

Theorem 226 (2 out of 3 property for flat maps) Let A rarr B and B rarr C be rings map withB rarr C faithfully flat Then

Ararr B rarr C is flatlArrrArr Ararr B is flat

222 For schemes

Definition 227 A map of schemes f X rarr Y is

(1) flat at x if flowast OYf(x) rarr OXx is flat(2) faithfully flat at x if f is surjective and flowast OYf(x) rarr OXx is flat

The map f is flat (resp faithfully flat) if it is flat (resp faithfully flat) for all points of X

A more geometric picture is captured in the following result

Theorem 228 Let f X rarr Y be a morphism of schemes Then f is

bull flat if the pull-back functor flowast QCoh(Y )rarr QCoh(X) is exactbull faithfully flat if the pull-back functor flowast QCoh(Y )rarr QCoh(X) is exact and faithful12

By the flatness of the map flowast OYf(x) rarr OXx Corollary 221 and Theorem 225 it followsthat OYf(x) minusrarr OXx is injective

Theorem 229 Let f X rarr Y be a locally finitely presented flat morphism Then f is universallyopen mapping13

Given y isin Y let Xy be the fiber of f over y

Xy

X

f

Specκ(y) Y

12A functor F C rarr D is faithful if HomC(AB) minusrarr HomD(F (A) F (B))13ie it remains open under arbitrary base change


Theorem 2210 Let X and Y be locally Noetherian and f X rarr Y flat Then for x isin X andy = f(x)

codimXyx = codimXx minus codimY y

Theorem 2211 Let f X rarr Y be a faithfully flat morphism between irreducible varieties Then

dimXy = dimX minus dimY

for all y isin Y

If we relax the condition that the map is flat for every x isin X we may focus on the set of pointsfor which the map flowast OYf(x) rarr OXx It is called the flat locus of f The main theorem is

Theorem 2212 Let f X rarr Y be a morphism locally of finite type Then the flat locus of f isopen

Theorem 2213 Suppose f Ararr B is a local map of local noetherian rings such that

dimB = dimA+ dim(Bf(mA)B) dimB = dimA+ dim(Bf(mA)B)

and A is regular and B is Cohen-Macaulay Then f is automatically flat


3 Numerical invariants

Let X be a smooth curve over algebraically closed field k

Definition 301 For a projective curve X the genus of X is

g = dimkH1(XOX)

Definition 302 Let X be projective and E be a finite locally free OX-mod Then degree of E is

degE = χ(XE)minus rank(E) middot χ(XOX)

Definition 303 If f X rarr Y is dominant then it induces a morphism k(Y )rarr k(X) We call

deg f = [k(X) k(Y )]

the degree of f

Definition 304 If X sub PNk is a variety of dimension n then the leading coefficient of its Hilbert

polynomial is of the form dnx

n The number d is the degree of X If X is a curve then the degreeequals its leading coefficient

The above two definitions are closely related

Lemma 305 For a curve X sub Pnk

deg(X) = deg(OX(1))

Theorem 306 (Serre duality for curves) For a smooth curve X over algebraically closed field kwith canonical sheaf ωX we have

h0(L) sim= h1(ωX otimes Lminus1)


Theorem 307 (Riemann-Roch) For locally free E on X we have


In particular for line bundles14 we have

h0(L)minus h1(L) = deg(L) + 1minus g

or in terms of divisors

h0(OX(D))minus h1(OX(D)) = degD + 1minus g

Lastly combining with Serre duality this can also be written in the form

h0(L)minus h0(ωX otimes Lminus1) = deg(L) + 1minus g

31 Modules of differentials

14recall that L sim= OX(D) in the smooth case Then degL = degOX(D) = deg(D)


311 Over ringsLet A be a ring B be a A-algebra and M be an B-module

Definition 311 An A-derivation of B into M is a map d B rarrM which

(1) is additive(2) satisfies the Leibniz rule

D(bbprime) = bD(bprime) +D(b)bprime

(3) satisfies D(a) = 0 foralla isin A

Definition 312 The module of relative differential forms denoted (ΩBA d) to be theuniversal object through which every Aminusderivation dprime B rarr M factors through In other wordswe have a canonical bijection

HomB(ΩBAM) minusrarr DerA(BM)

Remark 311 We can explicitly construct ΩBA as the free B-module generated by 〈db b isin B〉quotioned by the submodule generated by

〈d(bbprime)minus b(dbprime)minus (db)bprime da d(b+ bprime)minus dbminus dbprime a isin A b bprime isin B〉

with the map d B rarr ΩBA by b 7rarr db

Proposition 311 (First exact sequence) Let Ararr B rarr C be ring homomorphisms Then thereis natural squence of C-modules

ΩBA otimesB C rarr ΩCA rarr ΩCB rarr 0

Proposition 312 (Second exact sequence) Let I C B and C = BI Then there is naturalsequnece of C- modules

II2 rarr ΩBA otimesB C rarr ΩCA rarr 0

b 7rarr botimes 1

Remark 312 Note that the map II2 rarr ΩBA is in fact linear

Lemma 313 For any monoid S sube B we have

ΩSminus1BAsim= Sminus1ΩBA

Theorem 314 Let Kk be a finitely generated field extension of k Then

dimk ΩKk ge trdeg(Kk)

Equality holds if and only if K is separable extension of a transcendental extension of k

Proposition 313 Let (Bm) be a local rings containing a field k sim= Bm Then the map (seeProposition 312)

mm2 rarr ΩBA otimesB k

is an isomorphism


312 Over topological spaces

Definition 315 Let X be a topological space and φ O1 rarr O2 homomorphism of sheaves ofrings Let F be an O2-module A O1-derivation into F is a map D O2 rarr F which

(1) is additive(2) satisfies D

∣∣Im(φ(O1))

equiv 0 and

(3) satisfies the Leibniz rule

D(fg) = fD(g) +D(f)g

for all local sections f and g

Denote DerO1(O2F) the set of all derivations into F

Definition 316 Let X and O1O2 be as above The module of differentials denoted ΩO2O1

is the object representing the functor F 7rarr DerO1(O2F) ie

HomO2(ΩO2O1F) = DerO1(O2F)

It comes with universal derivation d O2 rarr ΩO2O1

Lemma 317 The sheaf ΩO2O1is the sheaf associated to the presheaf

U 7rarr ΩO2(U)O1(U)

Lemma 318 For U sube X open there is a canonical isomorphism

ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)

compatible with the universal derivations

Lemma 319 Let f X rarr Y be a continuous map of topological spaces Then there is a canonicalidentification

fminus1ΩO2O1= Ωfminus1O2fminus1O1

compatible with universal derivations

Stalkwise we have the following

Lemma 3110 For any O1 rarr O2 as sheaves of rings of X and x isin X we have

ΩO2O1x = ΩO2xO1x

Proposition 314 Let X be topological space and OiOprimei be sheaves on X Any commutative

diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO

yields a canonical map

ΩO2O1rarr ΩOprime2O

prime1

d(f) 7rarr d(ϕ(f))

for any local section f on O2


Proposition 315 (Second exact sequence) Suppose we have surjective ϕ O2 rarr Oprime2 withker(ϕ) = I

I O2ϕ Oprime2

O1

OO gtgt

Then there is a canonical sequence of Oprime2-modules

II2 minusrarr ΩO2O1otimesO2 O

prime2 minusrarr ΩOprime2O1

minusrarr 0

f 7rarr f otimes 1

313 Over schemesLet f X rarr Y be a morphism of schemes Clearly all of the results from the previous subsectionscarry over to this setting Let us explicitly construct ΩXY

Definition 3111 Let ∆ X rarr X timesY X be the diagonal morphism Suppose I is the sheafof ideals corresponding to the closed subscheme ∆(X) sube X timesY X Then the sheaf of relativedifferentials on X is defined by

ΩXY = ∆lowast(II2)

Remark 313 Itrsquos not hard to show that the above definition is equivalent to covering X andY by affine opens such that on compatible f(V ) sub U with V = SpecB and U = SpecA we have

ΩVU = ΩBA and gluing the corresponding quasi-coherent sheaves ΩBA

Remark 314 If X = AnY = SpecOY [x1 xn] then ΩXY is the free rank n OX-modulegenerated by dx1 dxn

Proposition 316 (Base change) Given a pullback diagram

X timesY Y primef prime

gprime

Y prime

g

Xf

Y

it holds that

ΩXprimeY primesim= gprimelowast

(ΩXY

)

Proposition 317 (First exact sequence) Let Xfminusrarr Y rarr Z be morphism of schemes Then there

is a canonical sequence of sheaves on X

flowastΩYZ rarr ΩXZ rarr ΩXY rarr 0

Proposition 318 (Second exact sequence) Let f X rarr Y be a morphism of schemes and I bethe ideal sheaf of closed subscheme Z of X Then there is a canonical sequence of sheaves on Z

II2 rarr ΩXY otimes OZ rarr ΩZY rarr 0

Theorem 3112 Let X = PnA and Y = SpecA Then there is an exact sequence of sheaves

0rarr ΩPnAY rarr OPnA(minus1)n+1 rarr OPnA rarr 0


32 Hurwitzrsquos theoremWe study finite morphisms f X rarr Y and the relation between their canonical divisors

Definition 321 For any p isin X the ramification index ep is defined as follows If t isin OQ isa local parameter at Q = f(P ) then

ep = νp(t)

where t is identified with its image of f OQ rarr OP

Definition 322 The morphism f X rarr Y is separable if K(X) is separable extension ofK(Y )

Proposition 321 Let f X rarr Y be a finite separable morphism of curves Then

0 minusrarr flowastΩY minusrarr ΩX minusrarr ΩXY minusrarr 0

This shows that the sheaf ΩXY measures the difference of the canonical sheaves on X and Y Now let PQ t be as in Definition 321 Then dt is a generator for the free OQ-module ΩYQ anddu is a generator for the free OP -module ΩXP Then there exists g isin OP such that

flowastdt = g middot duWe denote g = dtdu



(ep minus 1)

33 Serre duality Normal and conormal sheavesLet X be projective scheme of dimension d over a field k

Definition 331 A dualizing sheaf on X is an object ωX isin Coh(OX) which represents thefunctor

F 7rarr Hd(XF)or

In other words HomOX (F ωX)αFsimminusminusrarr Hd(XF)or is bijection This yields a trace map

t = αωX (idωX ) Hd(XωX)rarr k

Proposition 331 If FG isin Coh(X) then HomOX

(FG

)is finite dimensional Therefore by

functoriality for an F isin Coh(X) the pairing

〈〉F HomOX (F ωX)timesHd(XF)rarr k

(ϕ ξ) 7rarr t(ϕ(ξ))

is perfect ie 〈〉F is a nondegenerate bilinear form

Theorem 332 Projective space Pdk has a dualizing sheaf

ωPdksim= OPdk

(minusdminus 1)

Proposition 332 (Serre duality) Let X = Pdk then

(1) Hd(XωX) sim= k(2) for every i ge 0 there is a functorial isomorphism



Lemma 333 Let X be a ringed space Let E be a finite locally free OX-module with dual Eor =HomOX (EOX) Then

(1) Ext iOX (F otimesOX EG) = Ext iOX (FEor otimesOX G) = Ext iOX (FG)otimesOX Eor and

(2) ExtiX(F otimesOX EG) = ExtiX(FEor otimesOX G)

In particular via local to global spectral sequence (Thm 142)

ExtiX(EG) = H i(XEor otimesOX G)

Proposition 333 Let X be a projective scheme over k of dimension d Then X has a dualizingsheaf ωX and in fact

ilowastωX sim= ExtnminusdPnk

(ilowastOX ωPnk

)

where Ximinusrarr Pnk is a closed immersion

Definition 334

bull A Noetherian local ring is Cohen-Macauley if and only if depth(A) = dimA15bull A locally Noetherian scheme X is called Cohen-Macauley if and only if OXx is Cohen-

Macauley for all x isin X

Remark 331 When X is Cohen-Macauley we get duality of the same type using an auxiliarycoherent sheaf ωX which coincides with ωX in the smooth case

Theorem 335 (Jacobian Criterion) A scheme X rarr Y is smooth at x isin X if and only if

SpecB = U

X

SpecA = V Y

with B = A[x1 xn](f1fc) such that

det(

(partfjpartxi

)1leijlec

)is a unit in B (In this case f1 fc form a regular sequence in A[x1 xn] for Noetherian A)

Let i Z minusrarr X be a closed immersion of Noetherian schemes with quasicoherent ideal sheafcalled sheaf of ideals

IZ = ker(OX


Recall that there is a correspondence

sheaves on X annihilated by IZ larrrarr sheaves on Z

Definition 336

bull The conormal sheaf of Z in X is

CZX = unique quasi-coherent sheaf with ilowastCZX = II2 = II2 = ilowast(I)

bull The normal sheaf of Z in X is

NZX = HomOZ (CZX OZ) = HomOZ (II2OZ)

15Recall that for local rings we always have depth(A) le dimA


Proposition 334 Suppose Ximinusrarr Pnk is smooth of dimension d over Spec(k) Then

0 minusrarr CXPnk minusrarr ilowastΩPnk minusrarr ΩXk minusrarr 0

Fact 1 (1st adjunction formula) Suppose Ximinusrarr Pnk is a regular immersion of codimension c = nminusd

ThenωX = Λc

(NXPnk

)otimes ilowastωPnk

and X is Cohen-Macauley so we have perfect duality in all degrees

Fact 2 (2nd adjunction formula) Suppose Ximinusrarr Pnk is smooth of dimension d over Spec(k) Then

Λd(ΩXk

) sim= Λnminusd(NXPnk

)otimes ilowast

(ΛnΩPnk

)

Proposition 335 Let X be a smooth projective scheme over k equidimensional of dimension dThen

ωX sim= Λd(ΩXk

) sim= ΩdXk

and we have perfect duality in all degrees

Proposition 336 If X = V (F1 Fc) is a global complete intersection of the surfaces V (F1) V (Fc) subePnk of degrees d1 dc then

ωX sim= OX(d1 + middot middot middot+ dc minus nminus 1)

Proof (sketch) Use that andtopE1oplusmiddot middot middotEn = andtopE1otimesmiddot middot middototimesandtopEn For line bundles Ei = Li we haveandtopLi = Li Since II2 = oplusci=1OX(minusdi) we derive

NXPnk =(ilowastII2

)or= ilowast

(oplusci=1 OX(minusdi)

)or= ilowast

(oplusci=1 OX(di)

)

By the above discussion then andtop(NXPnk

)= ilowastotimesci=1O(di) = ilowastO(d1+middot middot middot+dc) By first adjunction

formula

ωX = andtop(NXPnk

)otimes ilowastωPnk = ilowast

(OPnk (

sumdi)otimes OPnk (minusnminus 1)

)= OX(d1 + middot middot middot+ dc minus nminus 1)


4 Tangent spaces

41 Tangent space as derivations

Let X be a Kminusscheme where K is a field For x isin X let κ(x) = OXxmx be the residue field

Definition 411 The tangent space of X at x is

TXks = DerK(OXx κ(x))

Proposition 411 There is a canonical isomorphism

TXks = DerK(OXx κ(x))rarr HomK(mxm2xK)

Suppose X has a k-structure and x isin X(k) Then OXx has a natural k-structure and it descendsto a k-structure on the residue field κ(x) = OXxmx This structure is isomorphic to

Homk(mxm2x k)

For f isin OXx set (df)x to be f minus f(x) (mod m2x)

Definition 412 The tangent vector X isin TXks corresponding to h mxm2x rarr κ(x) is

Xf = h((df)x

)

Let α X rarr Y be a morphism of K-schemes which yields a map

α Oα(x) rarr OXx

Definition 413 The differential (dα)x TXks rarr TYkα(x) is the map given by

(dα)x(X)(f) = X(α(f))

Lemma 414d(β α)x = (dβ)α(x) (dα)x

Lemma 415 If X = X1 timesK X2 then TXkx = TX1kx oplus TX2kx

42 Tangent space via dual numbersLet X be a S-scheme with as structure given by π X rarr S

In this section we derive an alternative definition of the tangent space due to Grothendieckwhich relies on maps from the dual numbers ring

Definition 421 (Dual numbers) For any ring R the dual numbers over R is the ring R[ε](ε2)

Let x isin X and s = π(x) Consider

Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)

with the curved arrow being the inclusion of a point morphism from Spec(κ(x)) to X

Definition 422 The set of dotted arrows in (421) is called the tangent space of X over Sat x and is denoted by TXSx

Lemma 423 The set TXSx defined above has a canonical κ(x)-vector space structure

The definition suggests the following


Definition 424 The cotangent space of X over S at x is

(TXSx)lowast = ΩXSx otimesOXx κ(x)

Indeed this is justified by the following

Lemma 425 Let π X rarr S be a morphism of schemes and x isin X There is a canonicalisomorphism

TXSx = HomOXx(ΩXSx κ(x))

of κ(x)-vector spaces

Since we work over arbitrary scheme S Proposition 411 needs slight modification

Proposition 421 Let π X rarr S be a morphism of schemes x isin x and s = π(x) isin S Ifκ(x)κ(s) is separable then there is a canonical isomorphism

mx(m2x + msOXx) = ΩXSx otimesOXx κ(x)


5 Line bundles and divisor correspondence

For an irreducible scheme X we have a unique generic point η and Frac (OXη) = k(X) is thefunction field of X

Definition 501 A local Noetherian ring (Rm) is regular if the minimal number l(m) of gen-erators of m satisfies

l(m) = dimκ(mm2) = dimR

where dimR is the Krull dimension of R

Definition 502 A scheme X is regular in codimension 1 if every local ring OXx of dimensionone is regular

If we talk about divisors on a scheme X we will always assume that

The scheme X is Noetherian integral separated scheme which is regular in codimension 1

Definition 503 For a line bundle L over a smooth projective curve X and nonzero s isin Γ(XL)we set16

ordxL(s) = valuation of (ssx) ge 0 = maxn ssx isin mnx

where sx isin Lx is a local generator ie Lx = sx middot OXx The divisor associated to s is

divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]

Definition 504 Line bundle L on a curve X is very ample (relative to Y ) if

L = ilowastOPnY (1)

for some embedding i X rarr Pn It is called ample if for any coherent F the sheaf FotimesLotimesn is veryample for n 0 Over schemes of finite type over a notherian ring this is equivalent to Lotimesn beingvery ample for some n 0

Definition 505 Let X be a smooth curve and D =sumnx middotx isin Div(X) Then we have invertible

sheaf OX(D) given by

OX(D)(U) = g isin k(X) forallx isin U minus closed ordx(g) + nx ge 0Definition 506 Let X be a projective scheme For a finite locally free OX-mod we set

deg(E) = χ(XE)minus rank(E) middot χ(XOX)

Alternatively if E has rk(E) sections s1 sr isin Γ(XE) which are linearly independent at thegeneric point we get

0 minusrarr OotimesrXs1srminusminusminusminusminusrarr E minusrarr Q minusrarr 0

anddeg(E) = dimk Γ(XQ) = h0(Q)

For line bundle L we have

deg(L) = dimk Γ(XQ) =sum

xisinXminusclosed

ordxL(s)

Definition 507 An OXminusmodule F on a scheme X is said to be generated by (finitely many)global sections if there exist (finitely many) global sections sα such that each stalk Fx is generatedby the sα over the Ox

16note that we divide by a local parameter ndash for that reason we will have divOX (D)(1) = D


Definition 508 Given line bundle L a regular meromorphic section (rms) s is a nonzeroelement in Lη where η is the generic point

If s and sprime are two rms of L isin Pic(X) then sprime = fs for f isin k(X)times Thus we have a well-definedmap

c1 Pic(X)rarr Cl(X) =Div(X)

Prin(X)

L 7rarr class of divL(s)

(501)

Proposition 501 Let Z sub X be a proper closed subset and set U = X minus Z Then

a) there is a surjective homomorphism

Cl(X)rarr Cl(U)sumniYi 7rarr

sumni(Yi cap U)

b) if codim(ZX) ge 2 then Cl(X)rarr Cl(U) is an isomorphismc) if Z is irreducible subset of codimension 1 then there is an exact sequence

Zrarr Cl(X)rarr Cl(U)rarr 0

1 7rarr 1 middot Z

Example 501 Let Y sub P2k be a curve of degree d Then

Cl(P2k minus Y ) = ZdZ

Proposition 502Cl(X) = Cl(X times A1)

Fact 3 If X is smooth projective curve then for all f isin k(X)times we have

deg(divX(f)) = 0

Here are a few results showing the connection between line bundles and divisors

Lemma 509 If s is a rms of L and D = div(L) then

OX(D)middotsminusminusrarr L

is an isomorphism

Lemma 5010 If X is projective smooth curve and D isin Div(X) then

deg(OX(D)) = χ(OX(D))minus χ(OX)

equals deg(D)

As a consequence of (3) we see that the degree map factors through in (501) so that

deg Cl(X)rarr Zwith deg(L) = deg(c1(L)) and so deg(L1 otimes L2) = deg(L1) + deg(L2) We have a bijection

Pic(X)larrrarr Cl(X)

L 7minusrarr c1(L)

OX(D)larrminus [ D

For every affine open subset U = Spec(A) of a scheme X set K(U) to be the localization ofA with respect to the set of non-zero divisors Let K be the sheaf associated to the presheafU 7rarr K(U)



fi isin Γ(UiKlowast)


∣∣Ui

) In this case we define associate subscheme of codimension 1 to be the closed




6 Computations of grd

For a nonsingular projective variety X over an algebraically closed field k we relate the globalsections of a line bundle to effective divisors on the variety In this case for every L isin Pic(X)Γ(XL) is a finite dimensional vector space over k

Definition 601 Let L isin Pic(X) and 0 6= s isin Γ(XL) The following effective divisor D = (s)0

is called the divisor of zeros of s Over U sub X where L is trivial we have

ϕ L∣∣U

simminusrarr O∣∣U

Then ϕ(s) isin Γ(UO∣∣U

) and Uϕ(s) determines an effective Cartier divisor D on X

Proposition 601 Let X be a nonsingular projective variety over k = k Let D isin Div(X) anddenote L = OX(D) Then

(1) for each s isin Γ(XL) the divisor of zeros (s)0 is an effective divisor linearly equiv to D(2) every effective divisor Dprime linearly equivalent to D is of the form (s)0 for some s isin Γ(XL)(3) two sections s sprime isin Γ(XL) have the same divisor of zeros iff sprime = λs for λ isin ktimes

Thus |D| is in 1-to-1 correspondence with(Γ(XL)minus 0

)ktimes

A linear system d is a subset of |D| corresponding to V sube Γ(XL) given by

V = s isin Γ(XL) (s)0 isin d cup 0Remark Theorem 611 below can be rephrased in terms of linear systems To give a morphismϕ X rarr Pnk is equivalent to giving a linear system d on X without base points and elementss0 sn spanning the vector space V




Remark 601 If (L V ) is a grd but not bpf let D sub X is the largest effective divisor such thatall sections of V vanish on D Then (L(minusD) V ) is grdminusdeg(D) and bpf

Definition 603 The gonality of a curve X equals the smallest d isin Z for which X admits a g1d

61 Separating points and tangent vectorsLet X be a variety L isin Pic(X) and suppose s0 sn isin H0(XL) generate L ie at least one is

nonzero at a given closed point x isin X Then

ϕ = ϕL(s0sn) X minusrarr Pn

x 7rarr [s0(x) middot middot middot sn(x)]

satisfies ϕlowastO(1) = L This construction in fact can be taken as a definition of projective space viardquofunctor of pointsrdquo philosophy

Theorem 611 Let A be a ring and X a scheme over A

a) If ϕ Ararr PnA is an A-morphism then ϕlowastO(1) is an invertible sheaf on X globally generated bysi = ϕlowast(xi)


b) Conversely if L is an invertible sheaf on X globally generated by si then there exists uniqueA-morphism ϕ Ararr PnA such that L = ϕlowast(O(1)) and si = ϕlowast(xi)

As a consequence one can show that

Aut Pnk = PGL(n k)

Example 611 Consider a projective curve X sube P3 which is the image of

P1 rarr P3

[u t] 7rarr [u3 u2t ut2 t3]

Since X sim= P1 all line bundles are of the form OP1(d) for some d Clearly ilowastxo = u3 isin OP1(3) so themap comes from a linear system d of degree 4 corresponding to V sub Γ(P1OP1(3)) FurthermoreX is not contained in any hyperplane so

Γ(P3O(1)) minusrarr Γ(X ilowastO(1))

is injective Since dimV = Γ(P1OP1(3)) = 4 it follows that d is complete

We now answer the question when such morphism is an immersion Note that for a projective Xϕ is immersion if and only if it is a closed immersion

Theorem 612 Let V = span(s0 sn) sube Γ(XL) and let L = OX(D) Then ϕ is immersionif and only if

(A) (separating points) forallx y isin X- closed there exists s isin V st s vanishes at x but not at y Inother words x is not base point of |D minus y|

(B) (separating tangent vectors)17 forallx isin X- closed the map

s isin V s isin mxLx rarr mxLxm2xLx = mxm

2x

is surjective Equivalently s isin V s isin mxLx spans mxLxm2xLx In other words x is not

base point of |D minus x|

For smooth curves X we may consider

Q V = koplusn+1 rarr H0(XL)

λ 7rarr sλ

For x y isin X- closed (possibly equal) we set V (minusxminus y) = Qminus1(H0(XL(minusxminus y))

) Then

ϕminus immersion lArrrArr forallx y isin X minus closed dimV (minusxminus y) = dimV minus 2

s0 sn globally generate LlArrrArr forallx isin X minus closed dimV (minusx) = dimV minus 1

As a consequence we have

Proposition 611 Let X be a smooth projective curve of genus g and L isin Pic(x) with degL ge2g + 1 Then

ϕL X rarr Pdminusgk

is a closed immersion

17this condition implies that the morphism Xϕminusrarr Pnk does not map nonzero tangent vectors to 0 and OPnϕ(x) rarr

OXx is surjective by Nakayamarsquos lemma


62 Commonly used factsFor any effective divisor D =

sumnx middot x and line bundle L on a smooth projective curve X we have

0 minusrarr L(minusD) minusrarr L minusrarroplus

xisinXminus closed

OXxmnxx minusrarr 0

where the cokernel term denoted OD is a skyscraper sheaf In particular

H0(XL(minusD)) = ker(H0(XL)rarr H0(XOD)

)

Lemma 621 If h0(L) gt 0 and x isin X is general then

h0(L(minusx)) = h0(L)minus 1

Lemma 622 Let X be a smooth projective curve and x isin X - closed Then

bull h0(OX(x)) = 1lArrrArr gx ge 1bull h0(OX(x)) = 2lArrrArr gx = 0

Corollary 621 If g ge 1 then ωX is globally generated ie |K| has no basepoints

Proof If not globally generated then h0(ωX(minusxprime)) = g for some closed xprime RR + duality showh0(OX(xprime)) = 2 which contradicts Lemma 622

Corollary 622 If g ge 1 then

x isin X minus closed minusrarr Pic1(X) = L isin Pic(X) degL = 1x 7minusrarr OX(x)

Lemma 623 (Criteria for bpf and ampleness) Let D isin Div(X) for a curve X Then

a) |D| is base point free if and only if for every point x isin X

h0(OX(D minus x)) = h0(OX(D))minus 1

b) D is very ample if and only if for every two points x y isin X (possibly the same) we have

h0(OX(D minus xminus y)) = h0(OX(D))minus 2

Remark 621 Note that in particular ample divisors are bpf

This lemma has few nice corollaries

Corollary 623 Let D be a divisor on a curve X of genus g

(1) if degD ge 2g then |D| has no basepoint(2) if degD ge 2g + 1 then D is very ample

Corollary 624 A divisor D on a curve X is ample if and only if degD gt 0


h0(L) le degL

2+ 1


(1) L sim= ωX (2) L sim= OX (3) X is hyperelliptic curve L sim= Notimesi for some 1 le i le g minus 2 where (N H0(N)) is g1

2

18such line bundles come from the so-called rdquospecialrdquo divisors


7 Curves

Fix an algebraically closed field k

Definition 701 A variety19 X is a separated reduced irreducible scheme of finite type overalgebraically closed field k

Theorem 702 Let k be an algebraically closed field There is a natural fully faithful functort Var(k)rarr Sch(k) from the categories of varieties over k to schemes over k given by

t Var(k)rarrSch(k)

V 7rarr

closed irreducible subsetsof X = sp

(t(V )

)and Ot(V )

given by restriction of thesheaf of regular functions

The image of t is the set of quasi-projective schemes In particular t(V ) is integral separatedscheme of finite type over k

Definition 703 For a variety X the function field of X denoted k(X) is

k(X) = Frac(OXη) = rational maps Xminusrarr A1k

where η is the generic point of X

Itrsquos not hard to see that

trdegk(k(X)

)= dim(X) = dim(U)

for any nonempty open U sub X

Definition 704 A variety X is normal if OXη is a normal domain ie integrally closed in itsfield of fractions k(X)

Definition 705 A curve is a variety of dimension 1

Fact 4 A curve X is normal if any of the following equivalent conditions holds

(1) X is smooth(2) OXx is a DVR for all closed x isin X(3) Γ(UOX) is a Dedekind domain for every nonempty open U sub X(4) ΩXk is invertible OX-module(5) Every closed point x isin X is an effective Cartier divisor

Fact 5 If f X rarr Y is a morphism of projective varieties with finite fibres over closed pointsthen f is a finite morphism

Fact 6 Any curve is either affine or projective

Lemma 706 For a projective variety Xk the global sections H0(XOX) is finite extension of k

Corollary 701 If k = k and Xk is projective then H0(XOX) = k

19Hartshorne calls this rdquoabstract varietyrdquo see [1 II Ch 4]


71 Maps between curvesWe begin with a fundamental fact asserting that for smooth curves a rational map is defined at

every point

Proposition 711 Let CK be a curve V sube PN a variety P isin C a smooth point and letϕ C rarr V be a rational map Then ϕ is regular at P In particular if C is smooth then ϕ is amorphism

Proof Write ϕ = [f0 fn] with fi isin K(C) the function field Pick an uniformizer t at P Thenfor n = miniisin[N ] ordP (fi) we have ordP (tminusnfi) ge 0 foralli isin [N ] and ordP (tminusnfj) = 0 Hence tminusnfjare regular and tminusnfj(P ) 6= 0 so ϕ is regular at P

Corollary 711 A genus g = 0 curve X is rational ie birational to P1 only if X P1

Proposition 712 Let CK be a curve and let t be a uniformizer at some non-singular pointP isin C Then K(C) is a finite separable extension of K(t)

Let CK be a smooth curve and f isin K(C) a function Then f produces rational map

f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =

[f(P ) 1] if f is regular at P

[1 0] if f has a pole at P

Theorem 711 Let ϕ C1 rarr C2 be a morphism of curves Then ϕ is either constant or surjective

Theorem 712 Let C1K and C2K be curves

a) Let ϕ C1 rarr C2 be non-constant map defined over K Then K(C1) is a finite extensionof ϕlowastK(C2)

b) For any inclusion i K(C1) rarr K(C2) there exists non-constant ϕ C1 rarr C2 with ϕlowast = ic) Let L sub K(C) be a subfield of finite index containing K Then there exists unique a smooth

curve C primeK and a non-constant ϕ C rarr C prime such that ϕlowastK(C prime) = L

Corollary 712 Let ϕ C1 rarr C2 be a degree 1 map between smooth curves Then ϕ is anisomorphism

We conclude with the following

Theorem 713 There is an antiequivalence of categoriessmooth projective curve andnonconstant polynomials

larrrarr

finitely generated field exten-sions Kk with trdegk(K) = 1and k-field maps

X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr

(k(X)larr flowastk(Y )

)

72 Curves and divisors

Lemma 721 [1 II Cor 516] Let A be a ring and Y be a closed subscheme of PrA Then thereis homogeneous ideal I sub S = A[x0 xr] such that Y = Proj(SI)


Proof Let Γlowast(F) = oplusnisinZΓ(XF(n)) where X = Proj(S) It holds that Γlowast(OX) S and for any

quasi-coherent F we have Γlowast(F) F Apply this to the ideal sheaf IY of Y in X

Lemma 722 Let X sub P2 be a plane curve Then

X = Proj(k[T0 T2 T2]I(X)

)

where I(X) = (F ) for some irreducible homogeneous polynomial F isin k[T0 T1 T2]

Therefore for plane curve X sub P2 we have

deg(X) = degOX(1) = degF

Definition 723 A smooth projective curve X is called hyperelliptic if there exists a degree 2morphism

X rarr P1k

Lemma 724 If g ge 2 then X has at most one g12 When X is hyperelliptic then the canonical

embedding X rarr Pgminus1k splits as

Xfminusrarr P1

k

x 7rarr(xxx)minusminusminusminusminusminusminusminusrarr Pgminus1k

where f is the unique degree 2 morphism f X rarr P2k

We want to see when the canonical map is a closed immersion

Theorem 725 Let X be a smooth projective curve of g ge 2 Then the following are equivalent

(1) X - hyperelliptic

(2) the canonical map ϕ = ϕωX X rarr Pgminus1k is NOT closed immersion

When both hold then ϕ is a double cover of a rational normal curve in Pgminus1k and ωk Lotimes(gminus1)

where (L V ) is g12

Therefore if X is not hyperelliptic we have a canonical embedding X rarr Pgminus1k of degree 2g minus 2

Under this embedding

ωX = ilowastOPgminus1k

= OX(1) =rArr degOX(n) = n(2g minus 2) (721)

From the exact sequence0rarr Irarr OP rarr OX rarr 0

by passing to long exact sequence and twisting we get

0rarr H0(P I(n))rarr H0(POP(n))rarr H0(POX(n))

Using (721) and Riemann-Roch we may compute dimkH0(POX(n)) and from there dimkH

0(P I(n))Recall

H0(Pd I(n)) = degree n forms whose zero set is a degrre n surface Q sube Pd containing XThus dimkH

0(P I(n)) gt 0 implies the existence of such surfacesFor singular X we may wish to rdquoresolverdquo the singularities We do so using the normalization

morphismν Xν rarr X

where Xν is normal ν is finite and birational hence ν induces isomorphism on function fields

Lemma 726 Let f X rarr Y be a nonconstant morphism of projective curves Then

degX(flowastL) = deg(f) middot degY (L)

for L isin Pic(Y )


Remark 721 Applying this lemma for f = ν of deg ν = 1 shows that νlowast preserves degrees ofline bundles

Lemma 727 (Base point free pencil trick) Let X be a smooth curve L isin Pic(X) and F

be a torsion-free OX-module Let two linearly independent sections s1 and s2 span the subspaceV sube H0(XL) Then the kernel of the cup-product (multiplication) map

V otimesH0(XF) minusrarr H0(XF otimes L)

equals H0(XF otimes Lminus1(B)) where B is the base locus of V

Lemma 728 Let f X rarr Y be a birational morphism of projective curves Then

gX = dimkH1(XOX) le dimkH

1(YOY ) = gY

with equality if and only if f is an isomorphism

Corollary 721 Let X rarr P2 be a curve of degree d Then

gXν le gX =(dminus 1)(dminus 2)

2with equality if and only if X is smooth

Lemma 729 Every quadric20 X sub Pn is birational to Pnminus1

73 Genus 0We have a complete description of these

Proposition 731 If gX = 0 then X P1k

Proof Let xprime isin X be a closed point Then h0(OX(xprime)

)= 2 Pick nonconstant f isin H0(XOX(xprime))

so that f has at most one pole of order 1 Then

f X rarr P1k

has degree 1 and so is an isomorphism by Corollary 712

74 Genus 1Since h0(L) is uniquely determined by Riemann-Roch whenever deg(L) ge 2g minus 2 and 21minus 2 = 0

we obtain

Proposition 741 If g = 1 and L is a line bundle then

h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0

Lemma 741 If g = 1 then the map

x isin X minus closed minusrarr Pic1(X)

x 7minusrarr OX(x)

from Corollary 622 is surjective hence bijective

20irreducible variety which is the zero locus of a quadratic homogeneous polynomial


Group law on elliptic curves

Given (EO) where E is a smooth projective curve and O isin E is a closed point Then we defineaddition

+E E(k)times E(k) minusrarr E(k)

(x y) 7minusrarr unique z st OE(z) OE(x+ y minus O)

Here E(k) = Mor(Spec kE) are the k-points of E See also [1 IV Example 137]

The j-invariant

Let X be an elliptic curve over k = k Fix P0 isin X Then the linear system |2P0| determines

f X rarr P1

of degree 2 with f(P0) =infin By Hurwitz theorem (Thm 323) we see that f is ramified at precisely4 points There is unique automorphism of P1 fixing infin and sending two of the branched points to0 and 1 Therefore we may assume the branched points are 0 1 λinfin Set

j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2

Theorem 742 Let k be algebraically closed field of characteristic 6= 2 Then

(1) two elliptic curves X and X prime over k are isomorphic if and only if j(X) = j(X prime)(2) every element of k is a j-invariant for some elliptic curve X over k

Remark 741 In other words k can be considered as a moduli space of curves of genus 1

75 Genus 2

Proposition 751 Let X be a curve of genus 2 Then ωX is base point free g12 and so X is

hyperelliptic

Proof Direct application of Riemann-Roch

76 Genus 3Let X be a non-hyperelliptic curve of genus 3 Then consider the canonical curve

ϕ = ϕωX X rarr P2

where ϕ is of degree 4 By Lemma 722 it follows that X V (F ) where F isin k[T0 T1 T2] isirreducible homogeneous polynomial of degree 4



where ϕ is of degree 6

Lemma 771 The image ϕ(X) is contained in a quadric Q sub P3 which is unique and irreducibleFurthermore ϕ(X) is a complete intersection of Q with a cubic surface in P3

Proof See [1 p 342]

Lemma 772 Suppose X is a smooth projective non-hyperelliptic curve of genus 4 Then either

a) X has exactly two g13 which are bpf and the canonical image lies on a smooth quadric or

b) X has exactly one bpf g13 and the canonical image lies on a singular irreducible quadric


78 Other results for curves

Theorem 781 (General position theorem) Let X sub Pr be a nondegenerate curve of degree d andH sube Pr be a general hyperplane Then

X capH = p1 pdwith pi pairwise distinct and any r of them being linearly independent

Theorem 782 (Bertinirsquos theorem) Let X be a nonsingular closed subvariety of Pnk where k = kThen there exists a hyperplane H not containing X such that H capX is regular at every point Infact if dimX ge 2 then H capX is an irreducible variety Furthermore the set of such hyperplanesforms a dense open subset of the complete system |H| considered as a projective space

Theorem 783 (Max Noether theorem) If C is a non-hyperelliptic curve of genus g ge 2 thenoplusdge0

H0(CωotimesdC )

is generated in degree 1


8 Surfaces

In this section we study surfaces via the tools of intersection theory

81 First approach to intersection theory

Definition 811 Let X be a surface and let C D be two divisors ie curves on X They meettransversally at P isin C capD if the local equations f g of C D generate the maximal ideal at PIn particular both C and D are nonsingular at P

Theorem 812 There is a unique pairing DivX timesDivX rarr Z denoted C middotD such that

(1) if C and D are nonsingular curves meeting transversally then C middotD = (C capD)(2) it is symmetric and additive(3) it descends to Pic(X)times Pic(X)

Lemma 813 Let C1 Cr be irreducible curves on the surface X and let D be a very ampledivisor Then almost all curves Dprime isin |D| are irreducible nonsingular and meet each Ci transver-sally

Proof Embed X into Pn using the ample divisor D Then apply Bertinirsquos theorem (Thm 782)simultaneously to X and the curves Ci to see that most Dprime isin |D| have nonsingular Dprime cap Ci iepoints with multiplicity one

Lemma 814 Let C be an irreducible nonsingular curve on X and D be any curve meeting Ctransversally Then

(C capD) = degC(OX(D)otimes OC

)Proof Recall that OX(minusD) is the ideal sheaf of D on X Tensoring by OC we get

0 minusrarr OX(minusD)otimes OC minusrarr OC minusrarr OCcapD minusrarr 0

where OCcapD is the scheme-theoretic intersection ie the fiber product C timesX D ThereforeOX(minusD)otimesOC is the invertible sheaf corresponding to C capD21 The intersection is transversal soits degree is (C capD)

Definition 815 Suppose C and D are curves with no common irreducible component Then forP isin C capD the intersection multiplicity (C middotD)P at P is

lengthOXPOXP (f g) = dimk OXP (f g)

where f and g are the local equations for C and D

Proposition 811 Suppose C and D are curves with no common irreducible component Then

C middotD =sum

PisinCcapD(C middotD)P

Proof Consider the same exact sequence


The scheme C capD has support at the points of C capD and at each p isin C capD its structure sheafequals OXP (f g) Therefore

dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P

21we always take the dual of the ideal sheaf


On the other hand by the exact sequence above

dimkH0(XOCcapD) = χ(OC)minus χ(OX(minusD)otimes OC)

Hence dimkH0(XOCcapD) only depends on the equivalence class of D and by symmetry of C

The claims follow by replacing C and D with differences of non-singular curves using Bertini-typeargument

Example 811 To compute the self-intersection number C2 since the ideal sheaf of C on X isI = OX(minusC) we can massage the formula in Lemma 814 to derive

C2 = degC Hom(II2OC) = degC NCX

the degree of the normal sheaf

Example 812

bull If X = P2 we have Pic(X) = Z generated by the class of line h Since any two lines arelinearly equivalent and two lines intersect at one point h2 = 1 Then C sim nh and D sim mhare two curves of degrees n and m respectively Then

C middotD = nm

yielding new proof of Bezoutrsquos lemmabull If X sube P2 is a nonsingular quadric we have Pic(X) = ZtimesZ generated by the class of linesh = (1 0) and k = (0 1) Then h2 = k2 = 0 and h middot k = 1 because two lines in the samefamily are skew and two of the opposite meet at a point Then if C and D are of types(a b) (aprime bprime) we have

C middotD = abprime + aprimeb

82 Numerical intersections

Lemma 821 Let k be a field Let X be a proper scheme over k Let F be a coherent OX-moduleLet L1 Lr be invertible OX-modules The map

(n1 nr) 7minusrarr χ(XF otimes Lotimesn11 otimes otimes Lotimesnrr )

is a numerical polynomial in n1 nr of total degree at most the dimension of the support of F

Definition 822 Let X be a proper scheme over k Let i Z rarr X be a closed subscheme ofdimension d Let L1 Ld isin Pic(X) We define the intersection number (L1 middot middot middotLd middotZ) as thecoefficient of n1 nd in the numerical polynomial

χ(X ilowastOZ otimes Lotimesn11 otimes otimes L

otimesndd ) = χ(ZLotimesn1

1 otimes otimes Lotimesndd |Z)

In the special case that L1 = = Ld = L we write (Ld middot Z)22

Definition 823 Let X be a proper scheme over a field k and F isin Coh(X) Let Zi sub X be theirreducible components of Supp(F) of dimension d with generic points ξi isin Zi The multiplicityof Zi in F is

mi = lengthOXξi(Fξi)

Properties of the intersection numbers

bull (L1 middot middot middot (Lprimei otimes Lprimeprimei ) middot middot middotLd middot Z) = (L1 middot middot middotLprimei middot middot middotLd middot Z) + (L1 middot middot middotLprimeprimei middot middot middotLd middot Z)

22the equality of Euler characteristics follows by the projection formula


bull Let f Y rarr X be a morphism of proper schemes over k and let Z sub Y be an integralclosed subscheme of dimension d Then

(flowastL1 middot middot middot flowastLd middot Z) = deg(f |Z Z rarr f(Z)

)(L1 middot middot middotLd middot f(Z))

where deg(f |Z Z rarr f(Z)

)is zero if dim f(Z) lt dimZ = d

bull If Z sube X is an integral closed subscheme with nonzero s isin Γ(ZL1

∣∣Z

) then

(L1 middot middot middotLd middot Z) = (L2 middot middot middotLd middot Z prime)

where Z prime is the vanishing scheme of s of codimension one in Zbull If X is a curve ie an effective Cartier divisor then

L middotX = degL

bull If L1 Ld are ample and dimZ = d then

(L1 middot middot middotLd middot Z) gt 0

Definition 824 Let X be a proper scheme over k with an ample line bundle L For any closedsubscheme the degree of Z with respect to L denoted degL(Z) is the intersection number(Ld middot Z) where d = dim(Z)

Lemma 825 Let Z sub X be a closed subscheme of dimension le 1 and L isin Pic(X) Then

(L middot Z) = deg(L|Z)

If L is ample then degL(Z) = deg(L|Z)

Lemma 826 Let f Y rarr X be a finite dominant morphism of proper varieties over k Let L

be an ample invertible OX-module Then

degflowastL(Y ) = deg(f) degL(X)

where deg(f) is as in Definition 303

Proposition 821 (Asymptotic Riemann-Roch) If L-ample on projective X of dimension d then

h0(Loplusn) = dimk Γ(XLoplusn) =degL(X)

dnd +O(ndminus1)

83 Divisors

Definition 831 Let X be a Noetherian scheme An effective Cartier divisor is a closedsubscheme D sub X such that the following equivalent conditions hold

(1) For every x isin D there exists an affine open neighbourhood Spec(A) = U sub S of x such thatU capD = Spec(A(f)) with f isin A a nonzerodivisor

(2) the ideal sheaf I sube OX of D is an invertible OX-module(3) the sheaf of ideal is locally generated by a single element ie forallx isin D the stalk Ix sube OXx

is generated by a single element f which is not a zero-divisor

If D1 and D2 are two effective Cartier divisors on X then we set D1 +D2 to be the effective Cartierdivisor whose local equations are the product of the local equations for D1 and D2

Fact 7 For D1 D2 as above

OX(D1 +D2) sim= OX(D1)otimes OX(D2)

1 7rarr 1otimes 1


Example 831 Let L isin Pic(X) and suppose s isin Γ(XL) is a regular global section ie OXmiddotsminusrarr L

is injective Then we have an effective Cartier divisor D = Z(s) the zero scheme of s which is

defined as follows On every local trivialization ϕ L∣∣U

sim=minusrarr OU set Z(s) cap U to be cut out by ϕ(s)

Definition 832 Let D sube X be an effective Cartier divisor Then set

OX(D) = Iotimesminus1 = HomOX (IOX)

which comes with a canonical global section 1 OX rarr OX(D) whose zero scheme is D

Remark 831 Note that OX(minusD) = I is the ideal sheaf of D

Theorem 833 For quasi-projective X any L isin Pic(X) is isomorphic to

OX(D1)otimes OX(D2)otimesminus1

for effective Cartier divisors D1 D2 sube X Moreover one can choose Di to avoid any finite subsetof X

Theorem 834 Let X be a smooth variety Then any reduced closed subscheme D sub X all ofwhose irreducible components have codimension 1 is an effective Cartier divisor

Proof OXx are UFDs

Definition 835 Let X be a variety

bull A prime divisor D is an irreducible reduced closed subscheme of codimension 1bull A Weil divisor is a formal Z - linear combinationsum

miDi

Definition 836 Let f isin k(X)times Then

divX(f) =sum

DsubXminusprime

ordD(f) middotD

where ordD(f) = order of vanishing of f along D = valuation of the DVR OXη on f

Definition 837 Let X be a variety L isin Pic(X) and s - nonzero meromorphic section of L Set

divL(s) =sum

DsubXminusprime

divLD(s) middotD

where divLD(s) = ordD(ssη)

We have the map

c1 Pic(X)rarr Cl(X) = Div(X)divX(f)fisink(X)times


for a nonzero meromorphic section s of L

Fact 8 The map c1 has the following properties

bull If Xminus normal then c1 is injectivebull If Xminus smooth then Pic(X) sim= Cl(X)bull If L = OX(D) where D sub X is effective Cartier divisor then

c1(L) =sum

mi middotDi

where Di sub D are irreducible components and mi is the multiplicity of Di in D


The intersection product of Definition 822 factors through Cl(X)

Lemma 838 Let C1 C2 sub X be distinct closed curves Then

C1 middot C2 =sum

xisinC1capC2

ex(CC prime)

whereex(C1 C2) = lengthOXx

(OXx(f1 f2)

)= dimk(OC1xf2OC2x)

and f1 f2 isin OXx are the local equations of C1 C2

Example 832

1)(O(n) middot O(m) middot P2

)= mn

2)(O(a b) middot O(c d) middot P1 times P1

)= ad+ bc

Theorem 839 (Bezout) Let C1 C2 sub P2k be distinct curves of degrees d1 d2 respectively Then

d1d2 =sum

xisinC1capC2

ex(C1 C2)

whereex(C1 C2) = lengthOP2x

(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)

and f1 f2 are the local equations near the point x


χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1






Theorem 8311 (Adjunction formula) If C sub X is a curve (closed not necessarily smooth)then

ωC sim= OX(C)otimesOX ωX∣∣C

If C is smooth then ωC sim= Ω1C and so

Ω1Csim= OX(C)otimesOX ωX

∣∣C

By taking the degrees of both sides we obtain

Corollary 831 (Adjunction formula) Let C and X as above Then

2g(C)minus 2 = (C +KX) middot Cwhere g(C) = h1(COC)

Example 833 For quadric X sube P3 we have Pic(X) = Zoplus

Z Then KX = (minus2minus2) Thus ifC is a curve of type (a b) then C minusK is of type (aminus 2 bminus 2) Hence

2g minus 2 = a(bminus 2) + (aminus 2)b =rArr g = (aminus 1)(bminus 1)


Lemma 8312 Let ϕ C1 rarr C2 be a non-constant (and so surjective) morphism of smoothprojective curves Let Γ sub C1 times C2 be the graph of ϕ Then

(1) (id ϕ) C1sim=minusminusrarr Γ is isomorphism

(2) Γ middot F1 = 1 where F1 = prminus11 (p1) for a closed point p1 isin C1

(3) Γ middot F2 = deg(ϕ) where F2 = prminus12 (p2) for a closed point p2 isin C2

(4) Γ middot Γ = deg(ϕ) middot degC2(TC2) = deg(ϕ)

(2minus 2g2

)

Proof For (2) and (3) note OC1timesC2(Fi) = prlowastiOCi(pi) so that

Γ middot Fi = degΓ(OC1timesC2(Fi)∣∣Γ)

= degC1(pullback of OCi(pi) by C1

(idϕ)minusminusminusminusrarr Γ minusrarr C1 times C2priminusminusrarr Ci)

= degC1

(idlowast OC1(p1) if i = 1

ϕlowastOC2(p2) if i = 2

)=

1 if i = 1

deg(ϕ) if i = 2

For (3) we have an adjuction short exact sequence

0 minusrarr II2 minusrarr Ω1C1timesC2

∣∣Γminusrarr Ω1

Γ minusrarr 0

and note ΩC1timesC2 = prlowast1ΩC1 oplus prlowast2ΩC2 Then

minusΓ middot Γ = degΓ(IΓI2Γ) = degΓ(prlowast1ΩC1

∣∣Γ) + degΓ(prlowast2ΩC2

∣∣Γ)minus degΓ(Ω1

∣∣Γ)

and the first and last cancel while

degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2

) = deg(ϕ)(2minus 2g2

)

84 AmplenessIn this subsection we assume X to be a smooth projective surface We say that H is (very) ample

if OX(H) is (very) ample in the sense of Definition 504

Theorem 841 Let H be a very ample divisor corresponding to an embedding X rarr PN Then

bull the degree of X in PN equals H2bull the degree of C in X equals C middotH for any curve C sub X

Remark 841 For a fixed very ample H CH plays the role of the degree of a divisor on a curve

Fact 9 Let π X rarr Y be a finite morphism of varieties Then

Lminus ample on Y =rArr πlowastLminus ample on X

Fact 10 Ample divisors on smooth projective surfaces have the following properties

bull H middot C gt 0 if Hminus ample and C sub X - curvebull Hminus ample lArrrArr mHminus very ample for m 0bull Hminus ample lArrrArr forallF isin Coh(X) H i(XF(mH)) = 0 for i gt 0 and m 0bull Hminus ample lArrrArr forallF isin Coh(X) F(mH) is globally generated

Lemma 842 If H is an ample divisor then for every divisor D with D middotH gt H middotKX = n0 wehave H2(XOX(D)) = 0

Lemma 843 If H is an ample divisor and D is such that D middot H gt 0 D2 gt 0 then nD iseffective for n 0


Definition 844 The divisors D1 and D2 are said to be numerically equivalent denotedD1 equiv D2 if and only if (D1 minusD2) middotD = 0 forallD isin Div(X) We set

Num(X) = Div(X)numerical equivalence

which induces a non-degenerate bilinear form

Num(X)timesNum(X)rarr Z

Theorem 845 (Theorem of the base) The group Num(X) is a finitely generated free abelian

Theorem 846 (Hodge index theorem) The signature of the quadratic form on Num(X) hasexactly one +1 Equivalently for ample H

D middotH = 0 D 6equiv 0 =rArr D2 lt 0

Theorem 847 (Nakai-Moishezon Criterion) Let X be a smooth projective surface

D minus amplelArrrArr

D2 gt 0

D middot C gt 0 for every curve C sub X

Definition 848 A divisor D is numerically effective (nef) if and only if D middot C ge 0 for allcurves C sub X The nef cone in Num(X)otimes R is

NEF = ξ isin Num(X)otimes R ξ middot C ge 0 for all curves C sub X

Theorem 849 (Kleiman) The ample cone is the interior of the nef cone

Proposition 841 (Kleiman) If D is nef then D2 ge 0

Definition 8410 Let C sub X be a curve passing through P isin X The multiplicity of C at P denoted mp(C) is given by

mP (C) = maxn f isin mnp

where f isin OXp is the local equation of C

Proposition 842 (Seshadrirsquos criterion) Suppose there exists ε gt 0 such that

D middot C ge ε middotmP (C) forallp isin C minus closed point

Then D is ample


9 Blow-ups

Let A be a Noetherian ring I an ideal of A Consider the graded algebra

A = Aoplus I oplus I2 oplus middot middot middotLet f1 fn be a system of generators of I Let ti isin I = A1 denote the element fi consideredas homogeneous element of degree 1 and not fi isin A which is of degree zero The surjectivehomomorphism of graded A-algebras

φ A[T1 Tn]rarr A

Ti 7rarr ti

shows A is a homogeneous A-algebra23 Note that if P isin A[x1 xn] is a homogeneous polynomialthen

P (t1 tn) = 0lArrrArr P (f1 fn) = 0

Definition 901 Let A be a Noetherian ring and I be an ideal of A The canonical morphism

Proj(A)rarr SpecA

is called blowing of SpecA along V (I)

Remark 901 By studying ker(φ) we may identify A(tk) with the subalgebra of Afk generated byfifk

A more general definition is

Definition 902 24 Let X be a smooth surface and P isin X be a closed point The blow up of Xat P is a projective birational morphism

π X rarr X

where X is another smooth surface The exceptional divisor

E = πminus1(P ) sub Xis a smooth projective curve of genus 0 ie E sim= P1

k

Definition 903 Let A be a Noetherian ring and I be an ideal of A Pick f isin I The affineblow up algebra is

A[I

f] = a

fna isin In sim

where afn sim

bfm hArr fN (fmaminus fnb) = 0 for some N 0

Example 901 Let A = k[x y] and I = (x y)CA Then

A[ Ix

]= k[yx] = k[t](y minus xt) and A

[Iy

]= k[xy] = k[s](xminus ys)

Fact 11 Let A I f be as above Then

bull f is a nonzero divisor in A[ If ]

bull fA[ If ] = I middotA[ If ]

bull Af sim=(A[ If ]

)f

23ie a quotient of A[T1 Tn] by a homogeneous ideal24this particular blow-up is called monoidal transformation in [1 ChV 3]


Corollary 901 Let U sube XZ Then

a) bminus1(U)rarr U is an isomorphismb) bminus1(I) middot O

Xis locally generated by a single nonzero divisor hence we get an effective Cartier

divisorc) E = bminus1(Z) scheme-theoretically

d) bminus1(U) is scheme-theoretically dense in Xe) O

X(1) sim= O

X(E) and this sheaf is relatively ample

Fact 12 For a curve C sub X passing through the point P

πlowastC = mp(C) middot E + C prime

where C prime is the strict transform of C and mp(C) is as in Definition 8410

Lemma 904 The self-intersection number of the exceptional curve is E middot E = minus1

Proof The normal sheaf NEX

is OE(minus1) sim= OP1k(minus1) the claim follows from Example 811

Remark 902 In fact any curve E sim= P1k inside a surface X with E2 = minus1 is obtained as

exceptional curve by blowing up a point in another surface X prime

Theorem 905 (Embedded resolution of singularities) Given a smooth surface X and a curveC sub X there exists a sequence of blow-ups at closed points Pi isin Xi

Xn rarr middot middot middotX1 rarr X0 = X

such that the strict transform Cn of C in Xn is smooth

901 Dominating by (normalized) blowupThe results of this subsections apply only to surfaces

Definition 906 A normalized blow up at a point p isin X is the composition

X prime minusrarr Blp(X) minusrarr X

where the first map is normalization and the later is a blow-up at p isin X

Theorem 907 (Version 1) Let f Y rarr X be a proper birational morphism of surfaces Thenthere exists a commutative diagram

Xn middot middot middot X1 X0

Y X

exist ν

f

where X0 rarr X is the normalization and each Xi+1 rarr Xi is a normalized blow up at a closedpoint

Theorem 908 (Version 2) Let f Y rarr X be a proper birational morphism of smooth surfacesThen there is a factorization of f of the form

Y = Xn rarr Xnminus1 rarr middot middot middot rarr X1 rarr X0 = X

where each Xi+1 rarr Xi is the blow up of a closed point Moreover

n = C sube Y minus irreducible curves such that f(C) = q isin X for some point q


Lemma 909 There exists a closed subscheme of dimension zero25 Z and a map

BlZ(X)rarr Y rarr X

Definition 9010 Let I sub OX be the ideal sheaf of Z = p1 pm For each i let

li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)

Then introduce the following invariant

inv(ZX) = (max li j lj = max li)

Proposition 901 In the notation of 9010 pick any pi with maximal li Let π X prime rarr X be theblow-up at pi Set

πminus1(I) middot OXprime = Iprime middot OXprime(minusdE)

with d-maximal Let Z prime to be the closed subscheme of X prime defined by Iprime Then

bull dim(Z prime) = 0bull inv(Z prime X prime) lt inv(ZX) in lexicographical order

Definition 9011 Let U sube X be an open subset of a variety X Then a U -admissible blow-up

π X rarr X is a blow up of X in a closed subscheme Z sube X with Z cap U = empty

Theorem 9012 (Zariski Main Theorem) Let f Y rarr X be a birational morphism of varietiesLet y isin X with image x isin X If OXx is normal and dimy f

minus1(x) = 0 then x is contained in themaximal open U sube X over which f is an isomorphism

Corollary 902 If f Y rarr X is as in 9012 and X is a normal surface then XY consists offinite number of closed points

91 Universal properties and intersection theory

Proposition 911 The blow up of a variety is a variety

Theorem 911 Let π X rarr X be the blow of a variety Z sub X Let f Y rarr X be a morphism

of varieties Let πprime Y rarr Y be blow up of fminus1(Z) in Y Then there exists unique f such that thefollowing diagram commutes

Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime

Furthemore Y rarr X timesX Y is the scheme-theoretic closure of Yfminus1(Z)

Proposition 912 (Properties of f)

bull If f is a closed immersion then f is a closed immersionbull If f is an open immersion then f is an open immersionbull If f is birational then f is birationalbull If f is smooth then f is smoothbull If f is proper then f is proper

25ie a collection of finitely many points


Proposition 913 (Intersection theory on X) The natural maps πlowast PicX rarr Pic X and Z rarrPic X via 1 7rarr 1 middot E give raise to an isomorphism

Pic X sim= PicX oplus ZWe also have


(2) if CD isin PicX then (πlowastC) middot (πlowastD) = C middotD(3) if C isin PicX then (πlowastC) middot E = 0(4) E2 = minus1



) = 0 then πlowast(X) isin Pic(X) and C sim= πlowastπlowastC(7) For n isin Z we have

πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E

92 Comparison with cohomology of the blow-upLet X(C) be a complex manifold of dimension 2 Then

Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)

H2(X(C)Z) = H2(X(C)Z)oplus Z[E]

H3(X(C)Z) = H3(X(C)Z)

Z = H4(X(C)Z) = H4(X(C)Z) = ZIn terms of the Hodge structure

hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise


10 Classification of surfaces

In this section we briefly summarize the Enriques-Kodaira classification

Definition 1001 A minimal surface is one which doesnrsquot have a (minus1) curves on it

Definition 1002 The Plurigenera are given by

Pn = dimCH0(XωotimesnX )

where X = Ω2X

Definition 1003 The canonical ring of X is given by

RX =oplusnge0

H0(XωotimesnX )

Remark 1001 Pnrsquos and RX are birational invariants

Definition 1004 The Kodaira dimension κ(X) is given by

κ(X) =

minusinfin Pn = 0 foralln ge 1

trdegC(RX)minus 1 otherwise

Remark 1002 One can think of κ(X) as the rate of growth of plurigenera

Proposition 1001 Let X be a projective surface over C Then

κ(X) isin minusinfin 0 1 2

1001 κ(X) =infinThere are two type of such surfaces

bull Rational surfaces P2 P1 times P1bull Ruled surface (birationally equivalent to) C times P1 where g(C) ge 126

Note that these surfaces have huge birational automorphism groups One can deduce the followingresult

Theorem 1005 (Luroth) If F is a field satisfying C sub F sub C(x y) and trdegC(F ) = 2 then

F sim= C(s t)

for some s t isin F

Geometrically this is the same as

Theorem 1006 (Luroth) If there exists dominant rational map Pminusrarr X then X is rational

1002 κ(X) = 0There are four types of such

bull Enrique surfacebull K3 surfacebull Abelian surfacebull Hyperelliptic surface

Definition 1007 A smooth surface X is a K3 surface if π1(X) = 0 and ωX = OX

26the minimal ones of this type can be written as bundles P(E)rarr C


1003 κ(X) = 1These are the properly elliptic surfaces

Definition 1008 An elliptic surface X is a morphism π X rarr B where X is a smoothprojective surfaces B is a smooth projective surface and the general fibre of π is an elliptic curve

Definition 1009 π X rarr B is relatively minimal if therersquos no (minus1)-curve if the fibre of π

1004 κ(X) = 2 For some n gt 0 ωotimesnX defines a morphism

ϕωotimesnX X rarr PN

which is birational onto its image

Proposition 1002 This map only contracts (-2)-curves


11 Riemann hypothesis for curves over finite fields

Set k = Fq and let X be a smooth projective curve over k such that Xk = X timesSpec(k) Spec(k) isirreducible ie X is geometrically irreducible

Definition 1101 The zeta function of X over k is

ZXk(s) =sum

DsubXminuseff Cartier divisor

qminus degk(D)s =prod

xisinXminusclosed

1

1minus qminus degk(x)s

where degk(D) is the degree of the morphism D rarr Spec(k)

Lemma 1102 For any d ge 1 there are finitely many x isin X - closed with degk(x) = d

Consider the exact sequence

0 minusrarr Pic(X) minusrarr Pic(X)degminusminusrarr Z

A priori we donrsquot know that the last map is surjective so we assume it lands on δZ for someδ gt 0 (which we show is 1)

Lemma 1103 Pic(X) is finite

Proof Note we have a bijection

Pic0(X)rarr Pic2gminus2(X)

L 7minusrarr Lotimes ωX

Now every N isin Pic2gminus2(X) has global section by RR so

Pic2gminus2(X) le D sube X deg(D) = 2g minus 2 ltinfin

by Lemma 1102

Set h = Pic(X) We have

ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1

q minus 1middot qminuss degk(L)

=sum

deg(L)le2gminus2 and δ| deg(L)

qh0(L) minus 1

q minus 1middot qminussdegk(L) +

sumdge2gminus2 and δ|deg(L)

h(qd+1minusq minus 1

)q minus 1

middot qminussd

(1101)

As a consequence

Corollary 1101 ZXk(qminuss) is a rational function of qminuss

Set t = qminuss and look at the pole at 1

limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)

After some base-change shenanigans we derive

Lemma 1104 It holds that δ = 1

The expression in (1101) yields


Theorem 1105

ZXk(t) =f(t)

(1minus t)(1minus qt)for f(t) isin Z[t] of degree 2g with f(0) = 1 and f(1) = h

Theorem 1106 (Functional equation)

ZXk(1

qt) = q1minusgt2minus2gZXk(t)

Proof The main idea is two split the formula for ZXk as in (1101) and do a change of variables

L = ωxotimes (Lprime)minus1 Then the verification reduces to algebraic manipulations plus Riemann-Roch

Corollary 1102 The polynomial f(t) in the numerator has 2g distinct roots w1 w2g

Corollary 11032gprodi=1

wi = qg

Corollary 1104 For all i isin [2g] there is j isin [2g] such that qwi

= wj

Definition 1107 The kprime-rational points of X are given by

X(kprime) = MorSpec(k)(Spec(kprime) X) = x isin X minus closed deg x = [kprime k]

Corollary 1105

X(k) = q + 1minus2gsumi=1

wi

Using the d log trick minusd log(1minus t) =sum

nge1tn

n we conclude

Corollary 1106

X(kk) = qn + 1minus2gsumi=1

wni

where kn is the unique finite field with [kn k] = n

Corollary 1107 For all n

qn + 1minus2gsumi=1

wni ge 0

Theorem 1108 (Prime Number theorem27) For all i = 1 2g we have

|wi| lt q

Proof For every ε gt 0 there exists n such that arg(wni ) isin [minusε ε] This shows |wi| le q If |wi| = qfor some i there are two possibilities

(1) wi = q is real and so wj = 1 by Corollary 1104(2) there are two of Galois conjugate

Both cases violate Corollary 1107

27reminiscent of the fact that ζ(s) has no zeros on lt(s) = 1


Theorem 1109 (Riemann hypothesis for curves over Fq) For all i = 1 2g

|wi| = q12

Proof Similarly to Theorem 1108 it suffices to show |wni | le Cqn2 In fact we show that

|X(k)minus q minus 1| le 2g middot q12

using Hodge index theorem


12 Abelian varieties

The theory of abelian varieties is very similar to that of elliptic curves - just replace Ewith A 1 with g (the dimension of A) and whenever E occurs twice replace one copywith the dual Aor of A

ndash James Milne Introduction to Shimura varieties

Primary sources for this section are [2] and [4]Suppose G is a an algebraic group of a finite type over k If k is a perfect field then the theory

splits into two very different cases

bull affine algebraic groups which are subgroups of GLn(k) for some nbull abelian varieties which are proper algebraic groups

Theorem 1201 (Chevalley) If G is an algebraic group over a perfect field k then there exists acanonical exact sequence in the category of algebraic groups

1rarr Gaff rarr Grarr Ararr 0

where Gaff is an affine algebraic group and A is a proper algebraic group both defined over k

121 Basic properties of abelian varieties For most of this chapter we restrict to abelianschemes over a field but for completeness we present the general definition

Definition 1211 An abelian scheme A over a scheme S is a smooth proper group scheme withconnected geometric fibers As a group scheme it is equipped with

1 a unit section eS S rarr A2 a multiplication morphism AtimesS Ararr A3 an inverse morphism Ararr A satisfying the standard axioms for abstract groups

Lemma 1212 Let X be an irreducible proper variety over k = k let Y be an irreducible varietyand y0 isin Y Suppose u X times Y rarr Z is a regular map such that u(X times y0) is a point Thenu(X times y) is a point for all y isin Y

Corollary 1211 Any abelian variety A is an abelian algebraic group

Proof Consider u AtimesArarr A given by u(a b) = aminus1ba and note that u(Atimes 1A) = 1A

Corollary 1212 Let A be abelian variety and G an algebraic group Then any regular mapu Ararr G satifying u(1A) = 1G is a group morphism

Proof Consider uprime A times A rarr G given by uprime(a b) = u(aminus1)u(b)u(aminus1b)minus1 and note that uprime(A times1A) = 1G

Theorem 1213 Any abelian variety is a commutative group and a projective variety28

Itrsquos easier if we can reduce to k = k due to the abundance of ponts That is accomplished bythe following result

Proposition 1211 Let kprimek be a field extension Let X be a proper k-scheme If Xkprime is projectiveover kprime then so is X over k In particular we may pick kprime = k

28Establishing projectivity is the hard part of the theorem It took nearly 10 years to prove The next few sectionswill describe the ideas that went into the proof


122 Complex toriLet V be a complex vector space of dimension n and Λ sube V a lattice The quotient Y = VΛis naturally equipped with a structure of compact complex manifolds and has an abelian groupstructure Note that we have canonical identification Λ sim= H1(XZ)


alternating r-forms andr Λrarr Z simminusrarr Hr(YZ)

Let π V rarr Y be the quotient morphism and pick a line bundle L on Y Since V is contractibleπlowastL sim= OV Clearly for any u isin Λ we have ulowastπlowastL sim= πlowastL ie u gives raise to an automorphismof OV given by an invertible holomorphic function eu isin Γ(VOtimesV ) Introduce eu(z) = e2πifu(z)which have to satisfy the cocycle condition

F (x y) = fy(z + x) + fx(z)minus fx+y(z) isin Z

Definition 1221 The Chern map c1 H1(YOtimesY ) rarr H2(YZ) sends L to c1(L) with corre-sponding E and2Λrarr Z given by

E(x y) = F (x y)minus F (y x)

The image of Chern class map is the Neron-Severi group NS(Y ) and the kernel is Pic0(Y )Alternating forms corresponding to c1(L) with ample L are called polarizations

Lemma 1222 The group NS(Y ) consists of alternating 2-forms E and2Λrarr Z such that

E(ix iy) = E(x y)

Let E and2Λrarr Z be of the type above It gives raise to a Hermitian form λ on V via

λ(x y) = E(ix y) + iE(x y)

Proposition 1222 Let V be a complex vector space and VR the underlying real vector spaceThere is a bijection

Hermitian forms on V simminusminusrarrskew-symmetric forms H

on VR satisfying H(ix iy) =H(x y)

λ 7minusrarr H = Im λ

Theorem 1223 (Appell-Humbert) Isomorphy classes of holomorphic line bundles on Y = VΛare in bijection with pairs (λ α) where

bull λ isin NS(Y ) is a Hermitian form on V with Imλ being integral on Λbull α Λrarr S1 is a map such that

α(x+ y)eiπIm(λ)(xy)α(x)α(y)

The line bundle L(λ α) corresponding to (λ α) is given by the Appell-Humbert cocycle

eu(z) = α(u)eπuλ(zu)+ 12πλ(uu)

Theorem 1224 The line bundle L(λ α) is ample if and only if the corresponding Hermitianform E and2Λrarr Z is positive definite In this case

dimH0(YL(λ α)) =radic

det(E)

Remark 1221 This implies that polarizations are precisely the positive definite forms in NS(Y )


In a suitable base for Λ a polarization λ and2Λrarr Z is represented by a matrix(0 DminusD 0

)

where D = (d1 dn) for some di gt 0 such d1|d2| middot middot middot |dn

Definition 1225 The tuple D = (d1 dn) is said to be the type of the polarization λ Apolarization of type (1 1) is called principal

Theorem 1226 Let Y = VΛ where Λ sube V sim= Cn is a lattice Then the following are equivalent

bull Y can be embedded into a projective spacebull There exists an algebraic variety X such that Xan sim= Y bull There exist n algebraically independent meromorphic functions on Y bull There exists a positive definite Hermitian form H V timesV rarr C such that Im H ΛtimesΛrarr C

is integral ie integer-valued

1221 Abelian varieties and CM-types

Definition 1227 An extension EQ is a CM field of degree 2g if it is imaginary quadraticextension of a totally real extension ie EFQ with EF an imaginary quadratic extension andFQ a totally real extension of degree g

Note that for a CM-field E we have Hom(EC) = 2g and the complex conjugation belongs tothe set

Definition 1228 A CM-type is a choice of Φ sube Hom(EC) such that Φ t Φ = Hom(EC)

Remark 1222 A CM-type yields an isomorphism E otimesQ R sim= CΦ

Theorem 1229 Let OE be the ring of integers of E Then CΦι(OE)

is an abelian varietywhere ι(x) = (σ(x))σisinΦ

123 Theorem of the cubeThe following theorem is pivotal to proving that abelian varieties are in fact projective It alsoproduces a myriad of corollaries that come in handy in computations

Theorem 1231 (Theorem of the Cube) Let X Y be proper varieties and Z be an arbitraryvariety Let x0 isin X y0 isin Y z0 isin Z Let L isin Pic(X times Y times Z) If L|x0timesYtimesZ L|Xtimesy0timesZ L|XtimesYtimesz0 are trivial then L is trivial

Corollary 1231 Let X Y Z be proper varieties Then every line bundle on X times Y times Z is ofthe form prlowast12L3 otimes prlowast23L1 otimes prlowast31L2

Corollary 1232 Let A be an abelian variety and X be an arbitrary variety Let f g h X rarr Abe three morphisms Then for any line bundle L on A

(f + g + h)lowastL sim= (f + g)lowastLotimes (g + h)lowastLotimes (f + h)lowastLotimes flowastLminus1 otimes glowastLminus1 otimes hlowastLminus1

Corollary 1233 Let L be a line bundle on A Then

[n]lowastL sim= Lotimes(n2+n)2 otimes [minus1]lowastLotimes(n2minusn)2 sim= Lotimesn2(Lotimes [minus1]lowastL)otimes

nminusn22

Corollary 1234 If L is symmetric29 then [n]lowastL sim= Ln2

29that is [minus1]lowastL sim= L


Corollary 1235 (Theorem of the square) For any x y isin A and L a line bundle on A we have

T lowastx+yLotimes L = τlowastxLotimes T lowastyLwhere

Ta Ararr A x 7rarr x+ a

Corollary 1236 The map

φL A(kprime)rarr Pic(A(kprime)) x 7rarr τlowastxLotimes Lminus1

is a group homomorphism for any L isin Pic(A) where kprimek is any field extension It also holds

φτlowastxL = φL and φL(minusx) = φLminus1(x)

for x isin A(kprime)

Remark 1231 Hence φL is an isogeny if and only if φLminus1 is an isogeny so it is an isogenywhen L is either ample or anti-ample

1231 Dual variety and projectivity

Theorem 1232 (Seesaw theorem) Suppose k is algebraically closed X is a proper variety overk and Z is an arbitrary variety over k Let L isin Pic(X times Z)Then

(1) Z0 = z isin Z L|Xtimeszis trivial is closed(2) There exists M isin Pic(Z0) on such that L|XtimesZ0

sim= prlowast2M

For a proper variety X we have an easy criterion for triviality

Lemma 1233 Let X be a proper variety Then L isin Pic(X) is trivial if and only if Γ(XL) 6= 0and Γ(XLminus1) 6= 0

By Corollary 1236 we have the group homomorphism

φL A(k)rarr Pic(Ak)

From φL1otimesL2 = φL1 otimes φL2 we obtain a homomorphism φ Pic(A)rarr Hom(A(k)Pic(Ak))

Definition 1234 The dual abelian variety A of A is Pic0(A) = ker(φ) In other wordsPic0(A) = L isin Pic(A) τlowastxL = L forallx isin A(k)

We thus obtain the following exact sequence

0rarr Pic0(A)rarr Pic(A)rarr Hom(A(k)Pic(Ak)) (1231)

The following result partly explains why it Pic0(A) is called rdquodualrdquo The rdquovarietyrdquo part will bejustified soon too

Lemma 1235 Let L isin Pic(A) Then L isin Pic0(A) if and only if mlowastL = prlowast1L otimes prlowast2L wherem AtimesArarr A is the group operation (ie addition)

Definition 1236 The Neron-Severi group is defined to be the cokernel of the map in (1231)ie

NS(A) = Pic(A)Pic0(A)

Remark 1232 Geometrically NS(A) is the group of line bundles on Ak up to algebraic equiva-lence where L1 and L2 on Ak are algebraically equivalent if there exists a connected k-scheme Spoints s1 s2 isin S(k) and a line bundle M on AS satisfying Nsi = (Li)K i = 1 2

We list a few deep results concerning NS(A)


Theorem 1237 The following is true

a) (Theorem of the Base) The group NS(Ak) is always finitely generated over Z In particular ithas finite torsion subgroup is finite

b) The torsion elements of NS(Ak) consist of line bundles L numerically equivalent to 0 ie for

all integral curves C sub Ak the intersection number C middot L = degC(L∣∣C

) vanishes

Theorem 1238 Let L = OA(D) for an effective divisor D on A The following are equivalent

(1) L is ample(2) K(L) = φminus1

L (0) = x isin A(k) τlowastxL = L is finite(3) H(D) = x isin A x+D = D is finite(4) Lotimes2 has no basepoint (ie OA otimes Γ(ALotimes2) Lotimes2) and the resulting k-morphism A rarr

P(Γ(ALotimes2)) is finite

Corollary 1237 Every abelian variety is projective

Proof Pick 0 isin U sube A an affine open and D = A U a divisor Then H(D) is closed as a

projection of mminus1(D) under A timesD mminusrarr A So H(D) is complete But H(D) sube U since 0 isin U Hence H(D) is a proper variety inside an affine variety thus is finite30 The claim follows fromTheorem 1238

Corollary 1238 The map [n] Ararr A is a finite flat surjection

The flatness follows by Miracle Flatness (Theorem 2213)

1232 Properties of the multiplication-by-n mapPlaying with the Euler characteristic and noting that [n] Ararr A is a dominant morphism of propervarieties we derive (a) of the following For (b) and (c) we examine the tangent map (d[n])0 at 0

Theorem 1239 Let A be an abelian variety of dimension g Then

(1) deg([n]) = n2g(2) [n] is separable if and only if p - n where p = char(k)(3) The inseparable degree of [p] is at least pg

Corollary 1239 Let A be an abelian variety of dimension g over a field k Then A(k) iscommutative and divisible The map [n] Ararr A is a surjective homomorphism and

ker([n])(k) =

(ZnZ)2g p = char(k) - n(ZnZ)i with all 1 le i le g possible p | n

124 Picard scheme dual abelian variety and polarizations

1241 Picard scheme and dual varietyLet AS be a projective abelian scheme over S The Picard functor PicAS SchS rarr Grp isgiven by

PicAS(T ) = (L ι) L a line bundle on AtimesS T ι elowastTLsim= OT a trivializationsim=

where eS T rarr AT is the unit section Note that PicAS(T ) is a commutative group with identity(OAT id) under otimes

Definition 1241 An isomorphims (L ι) sim= (L ιprime) on PicAS is an isomorphishm θ L sim= Lprime

preserving the trivializations ie

30because an affine variety X is complete only if dimX = 0 Otherwise it would admit a nonconstant regularfunction in O(X)


elowastTL elowastTLprime

OT

elowastT (θ)

ι ιprime

Definition 1242 A trivialization of L on AT along eT

Lemma 1243 Let T be a k-scheme and fT AT rarr T be the projection map Then OT =fTlowast(OAT ) Hence OtimesT = fTlowast(O

timesAT

) and upon taking global sections

Γ(TOtimesT ) = Γ(AT OtimesAT

)

Remark 1241 Grothendieck introduced the notion of rigidification specifically to address theproblem of units By the above arguments any two rigidifications differ by a unit Γ(TOtimesT )

For local rings (such as fields) we have the following assuaging result showing that studyingrigidified line bundles is the same as studying all

Proposition 1241 The projection

(L ι) on Asim= rarr Pic(A)sim=

is bijection

Remark 1242 We have the following sheaf condition Given (L ι) and (Lprime ιprime) on AT if thereexists a Zariski cover Tα of T such that these rigidified line bundles become isomorphic over ATαthen they are isomorphic over AT

This shows that PicAS(T ) is a Zariski sheaf on any T Thatrsquos false without the rigidificationSuch a sheaf property is certainly necessary for the functor to have any hope of being representable

Theorem 1244 (Grothendieck) 31 The functor PicXS is represented by a smooth separated(group) S-scheme which is locally of finite type over S If S = Spec(k) for a field k the connectedcomponent Pic0

Xk is quasiprojective and is projective if X is smooth

Definition 1245 For an abelian variety AS the dual abelian variety AS is the connectedcomponent Pic0

XS

Definition 1246 Pick T = PicAS then Id isin Hom(PicAS PicAS) corresponds to a canonical

pair (Puniv ιuniv) on AtimesS PicAS called the Poincare sheaf32

By functoriality any line bundle (L ι) is the pullback φlowast(Puniv ιuniv) via the map φ T rarr PicAScorresponding to (L ι)

(L ι) = φlowast(Puniv ιuniv)

(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words

31more generally itrsquos enough to assume that X is a geometrically reduced geometrically connected proper k-scheme with a rational point

32sometimes we call Poincare sheaf its restriction to the connected component AtimesS Pic0AS = AtimesS A


Theorem 1247 There is a canonical isomorphism

(1A times φ)lowastPunivsim= mlowastLotimes prlowast1Lotimes prlowast2L

We have the following suprisingly difficult theorem

Theorem 1248

dimHn(Atimes APuniv) =

0 n 6= g

1 n = g

Corollary 1241

χ(Puniv) = (minus1)g

The similarity with Pic0(A) of Definition 1234 is justified by the following

Theorem 1249 Let k be a field Then

Pic0Ak = Pic0(A)

Theorem 12410 Let L be an ample line bundle Then the map

φL A(k)rarr Pic0(Ak)

x 7rarr τlowastxLotimes Lminus1

is surjective

Combining the above theorem and Theorem 1238 we obtain the following

Corollary 1242 If L is ample then φL is surjective with finite kernel In particular dimA =dim A

Theorem 12411 The dual variety A = Pic0Ak is smooth Furthermore the tangent space at 0

is T0(A) = H1(AOA)

Theorem 12412 Let A be an abelian variety over k = k of dimension g then dimH1(AOA) = gand

andH1(AOA) rarr Hlowast(AOA) is an isomorphism Hence

andrH1(AOA) sim= Hr(AOA) andhr(A) =

(gr

)

Remark 1243 This is analogous to the canonical isomorphism

Hr(CnΛZ)rarr alternating r -forms andr Λrarr Z

where Λ sim= Z2n is a lattice

Definition 12413 Let f Ararr B be a map of abelian varieties Then we get a dual map

f B rarr A

which for any k-scheme T is the restriction from the pullback map

PicBk(T )rarr PicAk(T )

L 7rarr flowastTL


1242 PolarizationNote that φL = φLotimesL0 for any L0 isin Pic0(A) and so φL depends only on the image of L in NS(A)Furthermore we show that ampleness of L is determined by the class [L] isin NS(A)

Remark 1244 By Nakai-Moishezon criterion (Theorem 847) for geometrically integral pro-jective k-schemes X

L isin Pic(X) is ample lArrrArr L has positive degree on every integral curve in Xk

Hence ampleness of a line bundle on X depends only on the class [L] isin NS(k) ie it is invariantin connected families

Proposition 1242 For any line bundles L N on A with N isin Pic0Ak(k) we have

L is ample lArrrArr LotimesN is ample

Equivalently

L is ample lArrrArr (1A times φL)lowastPuniv is ample

Remark 1245 This shows that φL encodes information about the ampleness of L (via pullbackalong the Poincare sheaf)

Definition 12414 A polarization of an abelian variety A is an isogeny λ A rarr A such thatλk Ak rarr Ak is φL for some ample L isin Pic(Ak) The polarization λ is principal if it is an

isomorphism (equivalently deg λ = 133)

Definition 12415 For a proper curve X over k admiting a rational point x0 isin X(k) theJacobian variety is the variety J(X) = Pic0

Xk

In particular

J(X)(k) = L isin Pic(Xk) degL = 0and T0J(X) sim= H1(XOX)

Remark 1246 In the setup above Pic0Xk is indeed representable so the definition makes sense

If X = E is an elliptic curve the Jacobian J(E) equals E = E (elliptic curves are self-dual bycanonical choice of a principal polarization given by the bundle L = O(0))

Theorem 12416 The Jacobian J(X) is smooth and admits a canonical principal polarization

The following two gadgets play pivotal role in the proof of this result

Definition 12417 For any d ge 1 the Abel-Jacobi map AJd is given by

AJd Xd rarr J(X)

(x1 xd) 7rarr O(x1 + middot middot middotxd minus dx0)

Definition 12418 The theta divisor Θ is the divisor

Θ = L isin J(X) Γ(XL((g minus 1)x0)

)6= 0

Indeed one shows that (AJ1)lowast φΘ J(x)rarr J(X) is in fact the map minusId J(x)rarr J(X)

33see Theorem 1279 for easy criterion when this condition holds


1243 Duality for abelian schemes

Theorem 12419 Let f Ararr B be an isogeny of abelian varieties Then f B rarr A is also an

isogeny and ker f = ker f In fact

ker f(T ) = (L ι) on BT ιflowastTLsim= OAT

Since deg(f) = dim Γ(ker fOker f ) we obtain

Corollary 1243 If f is an isogeny then deg f = deg f

A line bundle L on Atimes S yields a morphism34

κL S rarr Pic(A)

s 7rarr [s 7rarr L∣∣Atimess]

If P = Puniv is the Poincare line bundle then κP = Id Denote κ = κσlowastP A rarr Â where

σ Atimes A sim= AtimesA is the flipping isomorphism We have the following nice factorization

Proposition 1243 Let L isin Pic(A) Then

AÂ

A

κ

φL φL

From Corollary 1243 we obtain deg κ = 1 and so

Corollary 1244 The homomorphism κ is an isomorphism HenceÂ = A and φL = φL

Definition 12420 A morphism λ Ararr A is called symmetric if λ = λ

Proposition 1244 Let f Ararr B be a morphism and L isin Pic(B) Then f φL f = φflowastL ie

Af

φflowastL

B

φL

A Bf

oo

125 Tate modules and `minusadic representation

Theorem 1251 For abelian varieties A B over k the group Hom(AB) is torsion-free In factHom(AB) is a finitely-generated free abelian group

Definition 1252 For ` 6= p = char(k) the ( `-adic) Tate module of A is given by

T`(A) = limlarrminusn

A[ln]

where the inverse limit is being taken with respect to the obvious maps

A[ln+1][l]minusrarr A[ln]

Set V`(A) = T`(A)otimesZ Q34hence line bundles on the product S timesA can be viewed rdquofamilies of line bundles on A parametrized by Srdquo


Remark 1251 The Tate module can be viewed as an analog of the (singular) homology groupH1(AanZ) Indeed over k = C we have

TÀ sim= H1(AanZ)otimesZ Z` and V` sim= H1(AanZ)otimesZ Q`

It is also dual to the l-adic cohomology as a Galois module

In light of Corollary 1239 we obtain the following

Lemma 1253 As a Z`-module the Tate module satisfies

T`(A) sim= (Z`)g times (Z`)gwhenever ` 6= char(k)

Note that Gal(kk)Aut(A[m]) since [m]P = 0 =rArr [m](σ(P )

)= 0 As the action of Gal(kk)

commutes with the multiplication maps it extends to action on T`(A)

Definition 1254 The `-adic representation of Gal(kk) is the map

ρ` Gal(kk)rarr Aut(T`(A)

)sube GL

(V`(A)

)given by the natural action of the Galois group on the projective system A[lm]mge1

Theorem 1255 Let A and B be two abelian varieties over k Then the natural map

Hom(AB)otimes Z` minusrarr Hom(T`(A) T`(B))

is injective

Clearly the image of the induced map lands in HomZ`(T`(X) T`(Y ))Gal(kk) We have the fol-lowing famous conjecture of Tate concerning the image

Conjecture 1 (Tate) If k is a field finitely generated over its prime field35 Then

Hom(AB)otimes Z` rarr HomZ`(T`(X) T`(Y ))Gal(kk)

is a bijection

Theorem 1256 The Tate conjecture is true for

bull finite fields (Tate)bull number fields (Faltings)

In the case when A = B we have a more explicit description

Theorem 1257

(1) For φ isin End(A)otimesQ`deg(φ) = det(T`(φ))

where det End(V`(A))rarr Q` is the determinant on the vector space V`(A) = TÀotimesQ`(2) For φ isin End(A) the characteristic polynomial P (n) = det(n middot Id minus φ) of T`(φ) has integer

coefficients

Proposition 1251 Let B be a finite dimensional simple Q-algebra with center K (which isnecessarily a field) Then there exist

a) a reduced norm N0BK B rarr K such that any norm form N B rarr Q is of the form

N = (NKQ N0BK)r for some r ge 0 and

35ie k = Fp(t1 tn) or k = Q(t1 tn)


b) a reduced trace Tr0BK B rarr K such that any trace form T B rarr Q is of the form T =

φ Tr0Bk for some linear map φ K rarr Q

Definition 1258 If P (x) = x2g + a1x2gminus1 + middot middot middot a2g is the characteristic polynomial of φ Then

bull Nφ = a2g = deg φ is the norm of φbull Trφ = minusa1 is the trace of φ

Suppose A is simple Let the division algebra B = End0(A) have center K Set [B K] = d2

and [K Q] = e for some integers d e Hence N0BQ is a polynomial of degree de In light of

Proposition 1251 since deg is a polynomial of degree 2g we obtain

Proposition 1252 If A is simple then de | 2g

Over characteristic zero using the Lefschetz principle we can make the result even sharper

Proposition 1253 If char(k) = 0 and A is simple then dimQ End0(A) = d2e | 2g

126 Poincare complete reducibility and Riemann-Roch for abelian varietiesHereafter set End(A) = End(A)otimesQ

1261 Poincare complete reducibility

Definition 1261 Let AV0k be the category of abelian varieties up to isogeny

bull objects abelian varieties over kbull morphisms elements of Hom(AB)otimesQ

Lemma 1262 Any isogeny f A rarr B is invertible in AV0k In fact f A rarr B is an

isomorphism in AV0k if and only if it is an isogeny

Theorem 1263 (Poincare compete reducibility) Let A sube B be an abelian subvariety Then thereexists C sube B such that Atimes C rarr B is an isogeny

Remark 1261 The same statement does not hold for algebraic tori

Corollary 1261 The category AV0k is semi-simple

Definition 1264 For an abelian variety A the degree function deg End(A)rarr Z is given by

deg(f) =

deg f f is isogeny

0 otherwise

Theorem 1265 The degree function deg extends to a polynomial End0(A)rarr Q of degree 2g

Remark 1262 This implies that End(A) is discrete in End0(A)

1262 Riemann-Roch for abelian varieties

Theorem 1266 (Riemann-Roch for abelian varieties) Let L isin Pic(A) Then

χ(Lotimesn) =dL middot ng

g

where dLg is the leading coefficient of the polynomial n 7rarr χ(Lotimesn) In other words χ(Lotimesn) ishomogeneous of degree g = dimA

Let Gk be a group scheme of finite type and Xk scheme of finite type equipped with trivialG-action


Definition 1267 A G-torsor (or principal G-bundle) P over X is a scheme P with a rightG-action together with a G-equivariant morphism π P rarr X such that the natural morphismP timesk Grarr P timesX P is an isomorphism

Lemma 1268 A G-torsor P over X is trivial if and only if P (X) = Hom(XP ) 6= empty

Proof Given a section s X rarr P the map (x g) 7rarr s(x)g is an isomorphism X timesG sim= P

Theorem 1269 Let G be finite π P rarr X be a G-torsor and X be proper Then for anyF isin Coh(X)

χ(πlowastF) = deg(π) middot χ(F)

Lemma 12610 Any L isin Pic(A) on an abelian variety A over k can be written as L = L1 otimesL2where L1 is symmetric and L2 isin Pic0(A)

Corollary 1262 The Neron-Sever group NS(Ak) is identified with the symmetric line bundleson A ie L isin PicA such that [minus1]lowastL sim= L

127 Weil pairing and Rosati involution Classification of End(A)

1271 Weil pairing

Let f Ararr A be an isogeny The duality ker f sim= ker f is given as follows at the level of k-pointsSuppose x isin ker f(k) and (L ι) isin ker f(k) By definition ι flowastL sim= OA and so τlowastx ι τlowastxf

lowastL rarrτlowastxOA

sim= OA Since τlowastxflowastL = flowastT lowastf(x)L = flowastL as f(x) = 0 we see that τlowastx ι is another isomorphism

flowastL sim= OA Thus τlowastx ι ιminus1 isin ktimes (see Lemma 1243) Denote this value by 〈xL〉f

Definition 1271 The Weil pairing is the map

〈middot middot〉n A[n]times A[n]rarr micron

(x (L ι)) 7rarr 〈xL〉[n]

associated to [n] Ararr A the multiplication-by-n isogeny

Lemma 1272 Let L isin A[m](k) and x isin A[nm](k) Then

〈xL〉nm = 〈nxL〉m

As a consequence we have the commutative diagram

A[`n]times A[`n] micro`n

A[`n+1]times A[`n+1]

[`]times[`]

OO

micro`n+1

`

OO

Hence we obtain a Weil pairing

〈middot middot〉ìnfin T`(A)times T`(A)rarr limlarrminusn

micro`n = Z`(1)

Remark 1271 In fact for arbitrary isogeny f Ararr B we have

〈T`(f)(x) y〉ìnfin = 〈x T`(f)(y)〉ìnfin

for any x isin T`(A) and y isin T`(B)


Theorem 1273 Let L isin Pic(A) The bilinear pairing

EL T`(A)times T`(A)1timesφLminusminusminusrarr T`(A)times T`(A)

〈middotmiddot〉ìnfinminusminusminusminusrarr Z`(1)

is skew-symmetric

Corollary 1271 Let f Ararr B be an isogeny and L isin Pic(B) Then

EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)

Proposition 1271 Let P be the Poincare line bundle on Atimes A Then

EP((x x) (y y)

)= 〈x y〉ìnfin minus 〈y x〉ìnfin

The following results shows that skew-symmetric pairings on T`(A) (over algebraically closedfields) arise from polarizations

Theorem 1274 Let φ Ararr A be a homomorphism Then the following are equivalent

a) φ is symmetric (ie φ = φ)b) Eφ is skew-symmetricc) 2φ = φL for some L isin Pic(A)d) Over k φ = φLprime for some Lprime

Combining this theorem with Corollary 1262 we obtain

Corollary 1272

bull We have

φL = 0lArrrArr Lk φM(x) for some x isin A(k) and an ample M

bull A line bundle L comes from Pic0Ak(k) if and only if φL Ararr A vanishes

bull The Neron-Severi group NS(Ak) rarr Hom(Ak Ak) can be identified with the symmetric

homomorphisms from Ak to Ak

Remark 1272 In other words the φ-construction detects when two line bundles lie in the samegeometric connected component of

(PicAk

)k

Lemma 1275 If L 6sim= OA and φL = 0 then Hr(AL) = 0 for all r

Remark 1273 Any non-trivial line bundle with vanishing φ has trivial cohomology groups Forline bundles with φ an isogeny it is known that the cohomology is supported in a single degree

1272 Rosati involution

Definition 1276 Let λ Ararr A be a polarization The Rosati involution associated to λ isthe anti-involution

(minus)prime End0(A)rarr End0(A)

φ 7rarr φprime = λminus1 φ λ

Lemma 1277 We haveEλ(T`(φ)(x) y

)= Eλ

(x T`(φ

prime)(y))

Theorem 1278 The Rosati involution is positive ie for 0 6= φ isin End0(A) Tr(φφprime) gt 0

Theorem 1279 For any ample L isin Pic(A) we have

deg(φL) = χ(L)2


1273 Classification of endomorphism algebras

Definition 12710 Let D be a finite dimensional division algebra with center K (which is a field)For a place v of K the local invariant of D is given by

invv(D) = [D otimesK Kv] isin Br(Kv) minusrarr QZ

We have

(1) Br(k) = 0 for k- algebraically closed(2) Br(R) sim= 1

2ZZ(3) For a p-adic field F there is an isomorphism Br(F ) sim= QZ

Remark 1274 Letrsquos recall the explicit construction of identification

invv Br(Kv)simminusrarr QZ

An element [D] isin Br(Kv) is represented by a central division algebra D with center Kv Let Lv bethe maximal subfield of D unramified over Kv Let Frob be the Frobenius automorphism of Lv Bythe Noether-Skolem theorem there is an element α isin D such that Frobx = αxαminus1 forallx isin Lv Weset

invv([D]) = ord(α) (mod Z)

which is independent of the choice of D36 and α

Theorem 12711 Producing a division algebra over a number field field K (up to isomorphism)is the same data as a family

(iv) isinoplus

vminusfinite

QZoplusoplusvminusreal

1

2ZZ

such thatsum

v iv = 0

Definition 12712 An Albert division algebra is a finite dimensional division algebra D withan anti-involution (minus)prime D rarr D such that Tr0

DQ is positive

Remark 1275 For abelian variety A the algebra D = End0(A) is an Albert division algebra inlight of Theorem 1278

Lemma 12713 Let D be an Albert division algebra with center K Let K0 = a isin K aprime = a37Then K0 is totally real and either

bull K = K0 orbull KK0 is a totally imaginary quadratic extension

Here is the desired complete classification due to Albert

Theorem 12714 (Albert) An Albert algebra D is one of the following types

bull Type I D = K = K0 is totally realbull Type II K = K0 is totally real D is a quaternion algebra over K such D otimesQ R sim=prod

KrarrRM2(R) with the anti-involution corresponding to transpose of matricesbull Type III K = K0 is totally real D is a quaternion algebra over K with the standard anti-

involution xprime = Tr0DK(x) minus x such that D otimesQ R sim=

prodKrarrRH where H is the Hamilton

quaternion algebra over R

36recall that [D] = [Dprime] if D otimesK Mn(K) sim= Dprime otimesK Mm(K) for some m and n37clearly [K K0] isin 1 2


bull Type IV [K K0] = 2 (K is a CM-field) D is a division algebra over K of dimension d2For every finite place v of K invvD+invσ(v)D = 0 where σ K rarr K is the automorphisminduced by the anti-involution on D Moreover if σ(v) = v then invv(D) = 0 In this casewe have D otimesQ R sim=

prodKrarrCsimMd(C) with the anti-involution corresponding to conjugate

transpose

We can list the numerical invariant of these four typesFor a simple abelian variety A over k set D = End0(A) Set [K0 Q] = e0 [K Q] = e and

[D K] = d2 Let D0 = x isin D x = xprime and η =dimQD0

dimQD

Proposition 1272 If L sube D0 is a subfield then [L Q] | g

Combining this result with Propositions 1252 and 1253 we obtain the following table

Type e d η char(k) = 0 char(k) gt 0I e0 1 1 e|g e | gII e0 2 34 2e | g 2e | gIII e0 2 14 2e | g e | gIV 2e0 d 12 e0d

2 | g e0d | g

Example 1271 Let E be an elliptic curve Then End0(E) is one of the following

bull Q (Type I)bull a quaternion algebra over Q ramified at infin (Type III when char(k) gt 0)bull an imaginary quadratic field (Type IV)

128 Abelian varieties over finite fields 38

Let A be an abelian variety of dimension g over k = Fq Let A(q) be the abelian variety39 whichlocally is defined by raising coefficients of each polynomial defining A by ai 7rarr aqi ie

A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k

Since Frobq = 1 we get a Frobenius morphism πA A rarr A(q) sim= A which can be explicitlydescribed as

πA Ararr A(q)

[x1 xn] 7rarr [xq1 xqn]

The Frobenius map is also characterized as follows

Proposition 1281 The Frobenius map is the unique way of attaching to every variety V overFq a regular map πV V rarr V such that

(1) for every regular map φ V rarrW φ πV = πW φ

38we have very heavily borrowed from the corresponding chapter in [2]39base changes preserve limits and coproducts Hence if A admits an algebraic structure defined in terms of finite

limits (such as being a group scheme) then so does A(1) Furthermore base change means preserves properties likebeing finite type finite presentation separated affine and so on


(2) πA1 is the map a 7rarr aq

Let PA be the characteristic polynomial of ΠA It is of degree 2g and has constant coefficientdeg πA = qg

Theorem 1281

(1) Q[πA] sube End0(A) is semisimple

(2) (Riemann hypothesis) Let ω be a root of PA Then |σ(ω)| = q12 for any embeddingσ Q(ω) rarr C

Definition 1282 A Weil q-number is an algebraic integer π such that for every embeddingσ Q(π) rarr C |σ(π)| = q12 Two Weil q-numbers π πprime are called conjugate if there exists anisomorphism

Q(π) sim= Q(πprime)

π 7rarr πprime

Denote the set of Weil q-numbers by W (q) and the conjugacy classes by W (q) sim

Immediate consequence of the Riemann hypothesis is the following

Corollary 1281 If A is a simple abelian variety40 then πA is a Weil q-number

Let Σ(AV0Fq) be the isomorphism classes of simple objects in AV0

Fq Then we have a well definedmap

Σ(AV0Fq)rarrW (q) simA 7rarr πA

We have the following amazing result

Theorem 1283 (Honda-Tate) The map Σ(AV0Fq)rarrW (q) sim is a bijection

Hence to produce an abelian variety over a finite field all we need is a Weil q-integerFor a Weil q-integer π

σ(π) middot σ(π) = q = σ(π) middot σ(qπ) for all σ Q[π] minusrarr C

Hence σ(qπ) = σ(π) and so

bull Q[π] is stable under complex conjugationbull complex conjugation sends π to qπ independently of the embedding σ

As a conclusion we obtain a description of all Weil q-integers

Proposition 1282 Let π be a Weil q-number Then it is one of the following three types

bull q = p2m π = plusmnpm and Q(π) = Q

bull q = p2m+1 π = plusmnradicp2m+1 Q(π) is totally real and [Q(π) Q] = 2

bull For any embedding σ Q(π) rarr C σ(π) 6isin R Q(π) is CM

Lemma 1284 If π πprime isin E are two Weil q-integers such that ordv(π) = ordv(πprime) for all v|p then

πprime = ζπ for some root of unity ζ in E

Theorem 1285 (Tate) The injective map

Hom(AB)otimes Z` rarr HomGal(kk)(TÀ T`B)

is a bijection

40so that Q[πA] is in fact a field


Over finite field Fq we have even more knowledge of the structure of End0(A)

Theorem 1286 The center of End0(A) is Q[πA] Every abelian variety over Fq is of CM-type

Assume that A is simple Let D = End0(A) and K = Q[πA] Then for v a place of K we have

invvD =

12 v realordv πAordv q

[Kv Qp] v | p0 otherwise

Example 1281

a) Q(π) = Q Then m is even π = plusmnradicq isin Q and PA(t) = (t plusmnradicq)2 Since A is of CM-type Dmust be a quaternion algebra over Q So invvD = 0 for v 6= infin p and invinfinD = 12 HenceinvpD = 12 and D = Qpinfin is the unique quaternion algebra over Q ramified only at infin p Weknow that d = 2 and g = 1 thus A is an elliptic curve The p-rank of A is zero since thedivision quaternion algebra D otimes Qp can not acts on Vpet(A) We say such an elliptic curve Ais supersingular

b) Q(π) is totally real and m is odd Then π = plusmnradicq D is the quaternion algebra over Q(radicp) ram-

ified at two real places We know that d = e = g = 2 thus A is an abelian surfaces When basechange to Fq2 PAotimesFq2 (t) = (tminus q)4 So AotimesFq2 is isogenous to the product of two supersingular

elliptic curvesc) Q(π) is an imaginary quadratic extension of Q Then PA is an irreducible quadratic polynomial

thus g = 1 and A is an elliptic curve Because e = 2 we find that d = 1 and D = K Thereare two cases1 p does not split in K then there is only one place over p Looking at the action of DotimesQp onTpet we see that A is a supersingular elliptic curve We claim that there exists some N suchthat πN isin Q or equivalently π2q is a root of unity Since π2 minus απ + q = 0 we know that|π2q|v = 1 for any v | ` Since π is a Weil q-number |π2q|infin = 1 By the product formulawe know that |π2q|p = 1 So the claim is proved By the first case Q(π) = Q Aotimes FqN is asupersingular elliptic curve

2 p splits in K Let v1 v2 be the two places over p Then A is an ordinary elliptic curve(ie its p-rank is 1)

Theorem 1287 Let E be an elliptic curve over Fq Then there are 3 possibilities

bull Q(π) = Q E is supersingular D = Qpinfinbull Q(π) is an imaginary quadratic field D = Q(π) and p does not split E is supersingularbull Q(π) is an imaginary quadratic field D = Q(π) and p splits E is ordinary

Theorem 1288 All supersingular elliptic curves over Fq are isogenous with an endomorphismalgebra D = Qpinfin

Theorem 1289 (Grothendieck) Let A be an abelian variety over k = k char(k) gt 0 If A is ofCM-type then A is isogenous to an abelian variety defined over a finite field

For abelian varieties this result can be strengthened

Theorem 12810 Let E be an elliptic curve over k = k char(k) gt 0 Then D = End0(E) = Qif and only if E cannot be defined over a finite field

We have a convenient way of determining if an elliptic curve is supersingular


Proposition 1283 Let Eλ y2 = x(x minus 1)(x minus λ) Assume char(k) = p 6= 2 Then Eλ issupersingular if and only if the Hasse polynomial vanishes ie

h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0

Corollary 1282 Over Fq there are only finitely many supersingular elliptic curves up to isomor-phism

1281 Neron models

Proposition 1284 Let R be a DVR AR be an abelian scheme and K = Frac(R) ThenEnd(A) sim= End(AK)

Definition 12811 (Neron model) Let A be an abelian variety over K v a discrete valuationof K and S = SpecOv A smooth separated scheme AS over S representing the functor T 7rarrA(T timesS SpecL) is called Neron model of A

Remark 1281 In particular the canonical map AS(S) rarr AK(K) is an isomorphism TheNrsquoeron model is unique up to unique isomorphism

Theorem 12812 The Neron model exists for any abelian variety

Remark 1282 The fiber of a Neron model over a closed point of Spec(Ov) is a smooth commu-tative algebraic group but need not be an abelian variety - it could be disconnected or a torus If itis an abelian variety we say that it has good reduction

1282 Abelian varieties over CM fieldsLet A be an abelian variety of dimension g

Definition 12813 An abelian variety (A ι) is of CM-type (EΦ) if ι E rarr End0(A) of degree2g such that T0(A) sim= CΦ as E otimesQ C-module We say that A is of CM-type by E

Remark 1283 Equivalently (A ι) is of CM-type (EΦ) if

Tr(ι(a)∣∣T0(A)

) =sumφisinΦ

φ(a) for all a isin E

We will also have H1(AC) sim= oplusφErarrCCφ where Cφ is one dimensional vector space with E-actinggiven by φ

Proposition 1285 If (A ι) is of CM-type by a field F then A is isogenous to a simple Anss ofCM-type If char(k) = 0 and A is simple and of CM-type then End0(A) = E is a field of degree2g over Q

Recall from Theorem 1229 that given a CM-field E and a CM-type (EΦ) then AΦ = CΦι(OE)

is an abelian variety

Theorem 12814 Let A be an abelian variety with CM type (EΦ) Then A is isogenous to AΦHence the CM-type is an invariant of an isogeny class of abelian varieties

Proposition 1286 Let A be an abelian variety over C with CM by E Then A is defined overQ In fact there is a unique model of A over Q

Theorem 12815 (Neron-Ogg-Shafarevich) For any abelian variety A of CM-type (EΦ) over anumber field k A has good reduction at p if and only if the inertia subgroup Ip rarr Gal(Qk) actstrivially on TÀ


Theorem 12816 Let A be an abelian of CM-type over a number field k Let p be a prime of kover p Then after a possible finite base change of k A has a good reduction at p

Lemma 12817 Let (A ι) be an abelian variety of CM-type (EΦ) over a number field k sube Cwith a good reduction (A ι) at P C Ok over Fq = OkP Then the Frobenius morphism πA of Alies in the image ι(E)

Thus given (A ι) and a prime of good reduction P we get a Weil q-integer πA isin E The followingresult is crucial for establishing the surjectivity of the Honda-Tate bijection (Theorem 1283)

Theorem 12818 (Shimura-Taniyama Formula) Assume that k is Galois over Q contains Eand that A is an abelian variety with CM-type (EΦ) over k with a good reduction at a place P | pof k Let v | p be a place of E Then

ordv(πA)

ordv(q)=

(Φ capHv)

Hv

where Hv = σ E rarr k σminus1(P) = pv

Remark 1284 Let lowast be the complex conjugation on Q[π] Then ππlowast = q and so

ordv(π) + ordv(πlowast) = ordv(q) (1281)

Combining this equality with ordv(πlowast) = ordvlowast(π) and Φ capHvlowast = Φ capHv yields

ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=

(Φ capHv) + (Φ capHvlowast)

Hv= 1

which is consistent with (1281) In fact the Shimura-Taniyama Formula is the only simple formulaconsistent with (1281)

129 Complex multiplication

1291 Class field theoryLet E be a number field Class field theory produces a continuous surjective reciprocity map

recE AtimesE rarr Gal(EabE)

such that for every finite extension LE we have the commutative diagram

EtimesAtimesE Gal(EabE)

EtimesAtimesENmLE(AtimesE) Gal(LE)

recE

σ 7rarrσ∣∣∣L

recLE

For convenience we use the reciprocal map artE(α) = recE(α)minus1

1292 Reflex field and norm of CM-typeLet (EΦ) be a CM-type

Definition 1291 The reflex field Elowast of (EΦ) is a subfield of Q satisfying any of the followingequivalent conditions

a) σ isin Gal(QQ) fixes Elowast if and only if σΦ = Φb) Elowast is the extension of Q generated by the elements

sumφisinΦ φ(a) a isin E


c) Elowast is the smallest subfield of Q admitting a vector space with E-action for which

Tr(a∣∣V

) =sumφisinΦ

φ(a)

Let V be a vector space satisfying condition c) which might be regarded as Elowast otimesQ E-space

Definition 1292 The reflex norm is the homomomorphism NΦlowast (Gm)ElowastQ rarr (Gm)EQsatisfying NΦlowast(a) = detE a

∣∣V

for all a isin Elowasttimes

Let (A ι) be an abelian variety of CM-type (EΦ) By Remark 1283 applied to T0(A) anyfield of definition of (A i) contains Elowast

1293 Main theorem of complex multiplicationAny homomorphishm σ k rarr kprime produces a functor V 7rarr σV of varieties over k to varieties overkprime

Let (A ι) be abelian variety of CM-type (EΦ) The map ι E rarr End0(A) yields a new pairσ((A ι)) = (σA σι) over Eprime where σi(a) = σ(ι(a))

Over C a map σ isin Aut(C) defines a homomorphism

σ Vf (A)rarr Vf (σA)

induced by C2g 3 P 7rarr σP isin C2g Here Vf (A) = Λ otimes Af where A(C) sim= C2gΛ as an algebraictorus

Theorem 1293 (Main theorem of CM) Let (A ι) be an abelian variety of CM-type (EΦ) overC and let σ isin Aut(CElowast) For any s isin AtimesElowastf such that artElowast(s) = σ

∣∣Elowastab

there exists a unique

E-linear rdquoisogenyrdquo α Ararr σA such that α(NΦlowast(s) middot x

)= σx for all x isin VfA


References

[1] R Hartshorne Algebraic Geometry (document) 166 19 721 74 77 24[2] C Li Abelian varities Live-TeXed notes for a course taught by Xinwen Zhu 12 38[3] Q Liu Algebraic Geometry and Arithmetic Curves[4] J S Milne Abelian Varieties (v200) Available at wwwjmilneorgmath 12[5] J H Silverman The Arithmetic of Elliptic Curves[6] The Stacks Project Authors Stacks Project httpstacksmathcolumbiaedu (document)[7] R Vakil Foundations of Algebraic Geometry 113 201



12 Operations on sheaves

13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem

33 Serre duality Normal and conormal sheaves

4 Tangent spaces


42 Tangent space via dual numbers



61 Separating points and tangent vectors

62 Commonly used facts

7 Curves

71 Maps between curves


73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups


92 Comparison with cohomology of the blow-up




121 Basic properties of abelian varieties

122 Complex tori

123 Theorem of the cube


125 Tate modules and -adic representation

126 Poincareacute complete reducibility and Riemann-Roch for abelian varieties


128 Abelian varieties over finite fields


References


Contents

1 Definitions and formulae 411 Most common definitions 412 Operations on sheaves 613 Dimension theory 714 Spectral sequences 715 Euler characteristic and polynomial 816 Computations of cohomology 917 Divisors 1118 Formulas 1119 Intersection theory after blow-up 122 Fiber product of schemes 1421 Properness and separatedness 1522 Flatness and faithful flatness 153 Numerical invariants 1831 Modules of differentials 1832 Hurwitzrsquos theorem 2233 Serre duality Normal and conormal sheaves 224 Tangent spaces 2541 Tangent space as derivations 2542 Tangent space via dual numbers 255 Line bundles and divisor correspondence 276 Computations of grd 3061 Separating points and tangent vectors 3062 Commonly used facts 327 Curves 3371 Maps between curves 3472 Curves and divisors 3473 Genus 0 3674 Genus 1 3675 Genus 2 3776 Genus 3 3777 Genus 4 3778 Other results for curves 388 Surfaces 3981 First approach to intersection theory 3982 Numerical intersections 4083 Divisors 4184 Ampleness 449 Blow-ups 4691 Universal properties and intersection theory 4892 Comparison with cohomology of the blow-up 4910 Classification of surfaces 5011 Riemann hypothesis for curves over finite fields 5212 Abelian varieties 55121 Basic properties of abelian varieties 55122 Complex tori 56






















X prime

X

f

Y prime Y






)









Remark 111


extension k of k










U cup V










F(fminus1(V )


Example 121






F(V )

Example 122


(fminus1F

)x

= Ff(x)






jF(V ) =



FU∣∣U







13 Dimension theory



(k[x1 xn](f)












p = capxisinXmx







Hp(XExtq(FG)

)rArr Extp+q

(FG

)



)







χ(XF) =infinsumi=0



d 7rarr χ(XF(d)


(suppF

)5


)= 0




of OPnk is(n+tt

)isin Q[t]





then X = Y 6





(n+ t

n

)minus(n+ tminus d

n

)





7 We have




)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1



2





Hq(PnROPnR

(d)) =


0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n



)=(n+dd

)if d ge 0





)= 0 otherwise


i0







∣∣∣i0ip+1




H0(UF) Γ(XF)



limminusrarrU






∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References






















X prime

X

f

Y prime Y






)









Remark 111


extension k of k










U cup V










F(fminus1(V )


Example 121






F(V )

Example 122


(fminus1F

)x

= Ff(x)






jF(V ) =



FU∣∣U







13 Dimension theory



(k[x1 xn](f)












p = capxisinXmx







Hp(XExtq(FG)

)rArr Extp+q

(FG

)



)







χ(XF) =infinsumi=0



d 7rarr χ(XF(d)


(suppF

)5


)= 0




of OPnk is(n+tt

)isin Q[t]





then X = Y 6





(n+ t

n

)minus(n+ tminus d

n

)





7 We have




)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1



2





Hq(PnROPnR

(d)) =


0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n



)=(n+dd

)if d ge 0





)= 0 otherwise


i0







∣∣∣i0ip+1




H0(UF) Γ(XF)



limminusrarrU






∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References









Remark 111


extension k of k










U cup V










F(fminus1(V )


Example 121






F(V )

Example 122


(fminus1F

)x

= Ff(x)






jF(V ) =



FU∣∣U







13 Dimension theory



(k[x1 xn](f)












p = capxisinXmx







Hp(XExtq(FG)

)rArr Extp+q

(FG

)



)







χ(XF) =infinsumi=0



d 7rarr χ(XF(d)


(suppF

)5


)= 0




of OPnk is(n+tt

)isin Q[t]





then X = Y 6





(n+ t

n

)minus(n+ tminus d

n

)





7 We have




)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1



2





Hq(PnROPnR

(d)) =


0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n



)=(n+dd

)if d ge 0





)= 0 otherwise


i0







∣∣∣i0ip+1




H0(UF) Γ(XF)



limminusrarrU






∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References








F(fminus1(V )


Example 121






F(V )

Example 122


(fminus1F

)x

= Ff(x)






jF(V ) =



FU∣∣U







13 Dimension theory



(k[x1 xn](f)












p = capxisinXmx







Hp(XExtq(FG)

)rArr Extp+q

(FG

)



)







χ(XF) =infinsumi=0



d 7rarr χ(XF(d)


(suppF

)5


)= 0




of OPnk is(n+tt

)isin Q[t]





then X = Y 6





(n+ t

n

)minus(n+ tminus d

n

)





7 We have




)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1



2





Hq(PnROPnR

(d)) =


0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n



)=(n+dd

)if d ge 0





)= 0 otherwise


i0







∣∣∣i0ip+1




H0(UF) Γ(XF)



limminusrarrU






∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References






13 Dimension theory



(k[x1 xn](f)












p = capxisinXmx







Hp(XExtq(FG)

)rArr Extp+q

(FG

)



)







χ(XF) =infinsumi=0



d 7rarr χ(XF(d)


(suppF

)5


)= 0




of OPnk is(n+tt

)isin Q[t]





then X = Y 6





(n+ t

n

)minus(n+ tminus d

n

)





7 We have




)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1



2





Hq(PnROPnR

(d)) =


0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n



)=(n+dd

)if d ge 0





)= 0 otherwise


i0







∣∣∣i0ip+1




H0(UF) Γ(XF)



limminusrarrU






∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References



Hp(XExtq(FG)

)rArr Extp+q

(FG

)



)







χ(XF) =infinsumi=0



d 7rarr χ(XF(d)


(suppF

)5


)= 0




of OPnk is(n+tt

)isin Q[t]





then X = Y 6





(n+ t

n

)minus(n+ tminus d

n

)





7 We have




)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1



2





Hq(PnROPnR

(d)) =


0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n



)=(n+dd

)if d ge 0





)= 0 otherwise


i0







∣∣∣i0ip+1




H0(UF) Γ(XF)



limminusrarrU






∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References




7 We have




)minus dimH0

(P2OP2(minusd)

)= 1minus 0 = 1



2





Hq(PnROPnR

(d)) =


0 if q 6= 0 n(1

T0TnR[ 1

T0 1

Tn])d

if q = n



)=(n+dd

)if d ge 0





)= 0 otherwise


i0







∣∣∣i0ip+1




H0(UF) Γ(XF)



limminusrarrU






∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References



∣∣V

) = 0 then






where












(PnRF(d)



0(PnRF(d)









17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References


17 Divisors



IZ = ker(OX







∣∣Ui





OX(minusD) = IY






18 Formulas










χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1














(ep minus 1)


h0(L) le degL

2+ 1



2








(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References


(4) E2 = minus1




πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References




Lemma 202 Let

X timesS Yq

p

Y supeWg

V sube Xf S supe U



Lemma 203 Let

X timesS Uq

p

U

i

Xf

S



∣∣fminus1(U)

)













AotimesB Sminus1B




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References




X timesS Sprime

B

Sprime S






is also separated






X

X prime

πoo Pn

S

~~

S






bull M is flat if















222 For schemes










Xy

X

f

Specκ(y) Y







for all y isin Y










g = dimkH1(XOX)





the degree of f






deg(X) = deg(OX(1))





























b 7rarr botimes 1









is an isomorphism





∣∣Im(φ(O1))

equiv 0 and












ΩO2O1

∣∣U

= Ω(O2

∣∣∣U

)(O1

∣∣∣U

)







ΩO2O1x = ΩO2xO1x


diagram

O2 ϕ Oprime2

O1

OO

Oprime1

OO



prime1

d(f) 7rarr d(ϕ(f))




I O2ϕ Oprime2

O1

OO gtgt




minusrarr 0

f 7rarr f otimes 1









gprime

Y prime

g

Xf

Y

it holds that


(ΩXY

)











ep = νp(t)









(ep minus 1)



F 7rarr Hd(XF)or




(FG







ωPdksim= OPdk

(minusdminus 1)












(ilowastOX ωPnk

)


Definition 334





SpecB = U

X

SpecA = V Y


det(

(partfjpartxi

)1leijlec



IZ = ker(OX




Definition 336










ThenωX = Λc

(NXPnk




Λd(ΩXk


)otimes ilowast

(ΛnΩPnk

)


ωX sim= Λd(ΩXk

) sim= ΩdXk





NXPnk =(ilowastII2

)or= ilowast


)or= ilowast

(oplusci=1 OX(di)

)



formula

ωX = andtop(NXPnk


(OPnk (




4 Tangent spaces








Homk(mxm2x k)



Xf = h((df)x

)


α Oα(x) rarr OXx


(dα)x(X)(f) = X(α(f))







Spec(κ(x))

++Spec(κ(x)[ε])

X

π

Spec(κ(s)) S

(421)



























divL(s) =sum

xisinXminusclosed

ordxL(s) middot [x]


L = ilowastOPnY (1)











xisinXminusclosed

ordxL(s)







Prin(X)


(501)




sumni(Yi cap U)



1 7rarr 1 middot Z





deg(divX(f)) = 0




is an isomorphism



equals deg(D)




L 7minusrarr c1(L)

OX(D)larrminus [ D






∣∣Ui









ϕ L∣∣U







)ktimes


















Aut Pnk = PGL(n k)


P1 rarr P3










2x





λ 7rarr sλ


) Then













xisinXminus closed

OXxmnxx minusrarr 0



)




















h0(L) le degL

2+ 1



2



7 Curves




t Var(k)rarrSch(k)

V 7rarr


(t(V )

)and Ot(V )







trdegk(k(X)

)= dim(X) = dim(U)













every point






f C rarr P1

P 7rarr [f(P ) 1]

where

f(P ) =











larrrarr


X 7minusrarr k(X)k(

Xfminusrarr Y

)7minusrarr


)








)





X rarr P1k



Xfminusrarr P1

k








where (L V ) is g12









0(P I(n))Recall







for L isin Pic(Y )









1(YOY ) = gY











f X rarr P1k



we obtain


h0(L) =

0 deg(L) lt 0

0 deg(L) = 0 L 6 OX

1 L OX

d deg(L) = d gt 0



x 7minusrarr OX(x)









The j-invariant


f X rarr P1


j(λ) = 28 (λ2 minus λ+ 1)3

λ2(λminus 1)2




75 Genus 2


hyperelliptic









Proof See [1 p 342]










H0(CωotimesdC )



8 Surfaces

















C middotD =sum





dimkH0(XOCcapD) =

sumPisinCcapD

(C middotD)P










Example 812


C middotD = nm
























∣∣Z

) then



L middotX = degL













dnd +O(ndminus1)

83 Divisors








1 7rarr 1otimes 1














Proof OXx are UFDs



miDi


divX(f) =sum

DsubXminusprime

ordD(f) middotD



divL(s) =sum

DsubXminusprime

divLD(s) middotD


We have the map






c1(L) =sum

mi middotDi





C1 middot C2 =sum

xisinC1capC2

ex(CC prime)


(OXx(f1 f2)

)= dimk(OC1xf2OC2x)


Example 832


)= mn


)= ad+ bc


d1d2 =sum

xisinC1capC2

ex(C1 C2)


(OP2x(f1 f2)

)= dimk(OC1xf2OC2x)



χ(XL) =1




χ(XOX(D)) =1

2D middotD minus 1


1










∣∣C













(2minus 2g2

)





= degC1



)=

1 if i = 1

deg(ϕ) if i = 2




Γ minusrarr 0





∣∣Γ)


degΓ(prlowast2ΩC2

∣∣Γ) = degC2

(ϕlowastΩ1C2


)






















D2 gt 0











Then D is ample


9 Blow-ups



φ A[T1 Tn]rarr A

Ti 7rarr ti




Proj(A)rarr SpecA





π X rarr X



k


A[I

f] = a

fna isin In sim

where afn sim



A[ Ix


[Iy






)f








X(1) sim= O



















Y X

exist ν

f








BlZ(X)rarr Y rarr X


li = lengthOXpi

(OXpiIpi

)= dimk(OZpi)
















Y Y sup fminus1(Z)

X X sup Z

π

exist f f

πprime













πlowastOX(nE) =

Iminusn n le 0

OX

n ge 0

(8) KX

= πlowastKX + E


Z = H0(X(C)Z) = H0(X(C)Z) = Z

H1(X(C)Z) = H3(X(C)Z)


H3(X(C)Z) = H3(X(C)Z)


hpq(X) = hpq(X) +

1 if p = q = 1

0 otherwise







where X = Ω2X


RX =oplusnge0

H0(XωotimesnX )



κ(X) =










F sim= C(s t)

for some s t isin F



















ZXk(s) =sum



xisinXminusclosed

1













by Lemma 1102


ZXk(qminuss) =

sumLisinPic(X)

qh0(L) minus 1


=sum


qh0(L) minus 1




)q minus 1

middot qminussd

(1101)

As a consequence



limtrarr1

(tminus 1)ZXk(t) =h

δ(q minus 1)





Theorem 1105

ZXk(t) =f(t)



ZXk(1






wi = qg


= wj



Corollary 1105


wi


nge1tn

n we conclude

Corollary 1106


wni



qn + 1minus2gsumi=1

wni ge 0


|wi| lt q







|wi| = q12




































E(ix iy) = E(x y)














det(E)




)




















nminusn22
















sim= prlowast2M




















) vanishes














ker([n])(k) =











OT

elowastT (θ)

ι ιprime



timesAT



)





is bijection






XS





(Puniv ιuniv)

AtimesS T1Atimesφ

AtimesS PicAS

In other words







Theorem 1248


0 n 6= g

1 n = g

Corollary 1241




Pic0Ak = Pic0(A)




is surjective




is T0(A) = H1(AOA)




(gr

)





f B rarr A



L 7rarr flowastTL








Equivalently






Xk

In particular







AJd Xd rarr J(X)




)6= 0











κL S rarr Pic(A)





AˆA

A

κ

φL φL





Af

φflowastL

B

φL

A Bf

oo





A[ln]
















)sube GL

(V`(A)




is injective




is a bijection




Theorem 1257



coefficients


























deg(f) =

deg f f is isogeny

0 otherwise






g












1271 Weil pairing














[`]times[`]

OO

micro`n+1

`

OO



micro`n = Z`(1)








is skew-symmetric


EflowastL(x y) = EL

(T`(f)(x) T`(f)(y)

)


EP((x x) (y y)






Corollary 1272

bull We have






(PicAk

)k








)= Eλ

(x T`(φ

prime)(y))



deg(φL) = χ(L)2





We have









(iv) isinoplus

vminusfinite


1

2ZZ

such thatsum

v iv = 0


DQ is positive















transpose



dimQD




2 | g e0d | g





A

φ

Frob

ampamp

πA

A(q)

A

φ

Spec kFrobq Spec k


πA Ararr A(q)










Theorem 1281





π 7rarr πprime






















is a bijection






invvD =



Example 1281
















h(λ) =

pminus12sum

k=0

(pminus12

k

)2

λk = 0


1281 Neron models









Tr(ι(a)∣∣T0(A)

) =sumφisinΦ














ordv(πA)

ordv(q)=

(Φ capHv)

Hv





ordv(πA)

ordv(q)+

ordv(πlowastA)

ordv(q)=


Hv= 1








recE


recLE








Tr(a∣∣V

) =sumφisinΦ

φ(a)



∣∣V









∣∣Elowastab





References





13 Dimension theory




17 Divisors

18 Formulas







32 Hurwitzs theorem


4 Tangent spaces







7 Curves



73 Genus 0

74 Genus 1

75 Genus 2

76 Genus 3

77 Genus 4


8 Surfaces



83 Divisors

84 Ampleness

9 Blow-ups







122 Complex tori








References

algebraic geometry notes - columbia universitystanislav/notes/algebraic... · 2018. 11. 25. ·...

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