diophantine geometry · 2017. 2. 6. · diophantine geometry travis dirle january 24, 2017. 2....

Diophantine Geometry

Diophantine Geometry

Travis Dirle

January 24, 2017

Contents

1 Preliminary: Algebraic Geometry Review 1

2 Height Functions 35

3 Rational Points on Abelian Varieties 37

4 Diophantine Approx. and Integral Points on Curves 39

5 Rational Points on Curves of Genus at Least 2 41

i

CONTENTS

ii

Chapter 1

Preliminary: Algebraic GeometryReview

We use the notation k for a perfect field, k for an algebraic closure of k, and Gkfor Gal(k/k).

Definition 1.0.1. Affine n-space (over k), which we denote by An or Ank , is theset

An = {(x1, . . . , xn) : xi ∈ k}.

The set of k-rational points of An is the set

An(k) = {(x1, . . . , xn) ∈ An : xi ∈ k}.

Or also as the set

An(k) = {(x1, . . . , xn) ∈ An : σ(xi) = xi for all σ ∈ Gk}.

Definition 1.0.2. Let I be an ideal in k[X1, . . . , Xn] = k[X]. We associate to Iits set of zeros,

Z(I) = {x ∈ An : P (x) = 0 for all P ∈ I}.

Similarly, to each subset S of An we associate the ideal of polynomials vanishingon S,

IS = {P ∈ k[X] : P (x) = 0 for all x ∈ S}.

Definition 1.0.3. An affine algebraic set S is a set of the form S = Z(I) forsome ideal I in k[X]. The set S is said to be defined over k if its ideal IS canbe generated by polynomials in k[X].

The hilbert basis theorem says that any ideal of polynomials is generated by afinite number of polynomials. Thus algebraic sets can always be written as the

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CHAPTER 1. PRELIMINARY: ALGEBRAIC GEOMETRY REVIEW

common zeros of a finite collection of polynomials.

If V is an algebraic set defined over k by some ideal I , then its set of k-rational points is defined by

V (k) = {x ∈ An(k) : P (x) = 0 for all P ∈ I}

= {x ∈ V : σ(x) = x for all σ ∈ Gk}.

It is also convenient to define

IV,k = {P ∈ k[X] : P (x) = 0 for all x ∈ V }.

Lemma 1.0.4. i) Let Vi be algebraic subsets of An. Then arbitrary intersection∩iVi and finite unions V1 ∪ · · · ∪ Vr are algebraic sets.

ii) If S1 ⊂ S2 ⊂ An, then IS1 ⊃ IS2 .iii) If I1 ⊂ I2 ⊂ k[X], then Z(I1) ⊃ Z(I2).iv) If V is an algebraic set, then Z(IV ) = V .v) If I is an ideal in k[X], then IZ(I) =

√I .

This says that there is a natural bijection between algebraic sets and reducedideals, that is, ideals that are equal to their own radical. Also, algebraic setssatisfy the axioms of the closed sets of a topology.

Theorem 1.0.5. (Hilbert’s Nullstellensatz) Let I be an ideal of the ring k[X1, . . . , Xn]and let P be a polynomial vanishing at every point in Z(I). Then there is an in-teger r ≥ 1 such that P r ∈ I .

Definition 1.0.6. The Zariski topology on An is the topology whose closed setsare algebraic sets. The Zariski topology on an algebraic set S is the topologyinduced by the inclusion S ⊂ An.

Definition 1.0.7. A nonempty subset Z of a topological space X is irreducibleif it cannot be written as the union of two proper closed subsets of Z.

An is irreducible. The Zariski topology is highly non-Hausdorff, any nonemptysubset of An is dense in An.

Definition 1.0.8. An affine variety is an irreducible algebraic subset (for theZariski topology) of some An.

Lemma 1.0.9. i) An algebraic set V is irreducible if and only if its ideal IV is aprime ideal.

ii) An algebraic set is a finite union of varieties. If we insist, as we may, thatnone of the varieties be contained in another one, then this decomposition isunique.

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Definition 1.0.10. The varieties in the decomposition ii) of an algebraic set arecalled the irreducible components of the algebraic set.

As an example, let P ∈ k[X1, . . . , Xn] be a polynomial and let V = Z(P ) bethe algebraic set defined by P . Suppose that P = Pm11 · · ·Pmrr is the decom-position of P into irreducible factors, and set Vi = Z(Pi). Then the Vi’s arethe irreducible components of V . Indeed, each Vi is a variety, V = ∪ri=1Vi, andVi 6⊂ Vj for i 6= j.

The algebra of polynomials in n variables is naturally associated to the affinespace An. When we restrict polynomial functions to an affine subvariety V , it isnatural to identify any two polynomials that give the same function on V . Thuswe are led to the following:

Definition 1.0.11. Let V be an affine subvariety of An. The affine coordinatering of V is

k[V ] = k[x1, . . . , xn]/IV .

This algebra completely characterizes the variety V .

Definition 1.0.12. Projective n-space Pn is the set of lines through the origin inAn+1. In symbols,

Pn ={(x0, . . . , xn) ∈ An+1 : some xi 6= 0}

∼=

An+1\{0}∼

,

where the equivalence relation ∼ is defined by

(x0, . . . , xn) ∼ (y0, . . . , yn)

⇔ (x0, . . . , xn) = λ(y0, . . . , yn) for some λ ∈ k∗.

If P ∈ Pn is the point representing the equivalence class of the (n + 1)-tuple(x0, . . . , xn), the xi’s are called the homogeneous/projective coordinates forthe point P . The set of k-rational points of Pn, denoted by Pn(k), is the setof lines through the origin in An+1 that are defined over k. This is the set ofpoints in Pn for which we can find some homogeneous coordinates in An+1(k).Equivalently these are the points (x0, . . . , xn) with the property that for anynonzero coordinate xj , all of the ratios xi/xj are in k.

The Galois group Gk acts on Pn by acting on the coordinates,

σ(P ) = (σ(x0), . . . , σ(xn)) for P = (x0, . . . , xn) ∈ Pn and σ ∈ Gk.

Then one can show that

Pn(k) = {P ∈ Pn : σ(P ) = P for all σ ∈ Gk}.

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Definition 1.0.13. The field of definition of a point P = (x0, . . . , xn) ∈ Pn isthe smallest extension of k over which P is rational, namely,

k(P ) = k(x0/xj, x1/xj, . . . , xn/xj) for any j with xj 6= 0.

Equivalently, k(P ) is determined by the property

Gal(k/k(P )) = {σ ∈ Gk : σ(P ) = P}.

Recall that a polynomial ideal is homogeneous if it is generated by homogeneouspolynomials, or alternatively, if the homogeneous components of any polyno-mial in the ideal are again in the ideal. If P is a homogeneous polynomial, then

P (x0, . . . , xn) = 0⇔ P (λx0, . . . , λxn) = 0 for all λ ∈ k∗.

Definition 1.0.14. A projective algebraic set is the set of zeros in Pn of a homo-geneous ideal in k[x0, . . . , xn]. The Zariski topology on Pn is defined by takingthe projective algebraic sets to be the closed sets, and the Zariski topology onan algebraic set is the topology induced from the Zariski on Pn. A projectivevariety is a projective irreducible algebraic set. It is said the be defined over kif its ideal can be generated by polynomials in k[x0, . . . , xn].

The correspondence between homogeneous ideals and projective algebraic setsis very similar to the affine one; the only difference is the existence of an irrel-evant ideal, I0 generated by x0, . . . , xn. Notice that I0 defines the empty subsetof Pn, and any homogeneous ideal different from k[x0, . . . , xn] is contained inI0. Let us define a saturated ideal as a homogeneous ideal I such that if xif ∈ Ifor all i = 0, . . . , n, then f ∈ I; clearly, the ideal of polynomials vanishing on aprojective algebraic set is saturated. More precisely, the map I 7→ Z(I) gives abijection between reduced saturated ideals and projective algebraic sets. Further,a projective algebraic set Z is a projective variety if and only if IZ is a (homoge-neous) prime ideal in k[x0, . . . , xn].

Definition 1.0.15. The homogeneous coordinate ring of a projective varietyV ⊂ Pn is the quotient

S(V ) = k[x0, . . . , xn]/IV .

Note that unlike the case of k[V ] for affine varieties, the elements of S(V ) do notdefine functions on a projective variety V . An even more important observationis that the homogeneous coordinate ring depends on the embedding of V in Pn,it is not an intrinsic invariant of V .

Definition 1.0.16. Let (x0, . . . , xn) be homogeneous coordinates on Pn. Thestandard (affine) open subset Ui is the complement of the hyperplane definedby xi = 0.

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Definition 1.0.17. A quasi-projective algebraic set is an open subset of a pro-jective algebraic set. A quasi-projective variety is an irreducible quasi-projectivealgebraic set.

Definition 1.0.18. LetX be a variety and x′ a point onX . A function f : X → kis regular at x′ if there exists an open affine neighborhood U ⊂ X of x′, sayU ⊂ An, and two polynomials P,Q ∈ k[x1, . . . , xn] such that Q(x′) 6= 0 andf(x) = P (x)/Q(x) for all x ∈ U . The function f is regular onX if it is regularat every point of X . The ring of regular functions on X is denoted by O(X).

Definition 1.0.19. Let x be a point on a variety X . The local ring of X at x isthe ring of functions that are regular at x, where we identify two such functionsif they coincide on some open neighborhood of x. This ring is denoted by Ox,Xor simply by Ox if no confusion is likely.

Definition 1.0.20. Let X be a variety and Y ⊂ X a subvariety. The local ringof X along Y denote byOY,X is the set of pairs (U, f), where U is an open subsetof X with U ∩ Y 6= ∅ and f ∈ O(U) is a regular function on U , and where weidentify two pairs (U1, f1) = (U2, f2) if f1 = f2 on U1 ∩ U2. The ring OY,X is alocal ring, its unique maximal ideal being given by

MY,X = {f ∈ OY,X : f(x) = 0 for all x ∈ Y }.

For example,O{x},X is just the local ring ofX at x, while the local ringOX,Xturns out to be a field.

Definition 1.0.21. Let X be a variety. The function field of X , denoted byk(X), is defined to be OX,X , the local ring of X along X . In other words, k(X)is the set of pairs (U, f), where U is an open subset of X and f is a regularfunction on U , subject to the identification (U1, f1) = (U2, f2) if f1 = f2 onU1 ∩ U2. (N.B. An element f of k(X) is not a function defined at every pointof X . Instead, f is a function that is defined at some point of X , and hence isdefined on a nonempty open set of points of X).

It is easy to check that k(X) is a field that containes every local ring OY,X of X ,and that for any subvariety Y ⊂ X , we haveOY,X/MY,X ∼= k(Y ). The functionfields of An and Pn are both equal to k(x1, . . . , xn).

Definition 1.0.22. A map φ : X → Y between varieties is a morphism if it iscontinuous, and if for every open set U ⊂ Y and every regular function f on U ,the function f ◦ φ is regular on φ−1(U). A map is regular at a point x if it is amorphism on some open neighborhood of x.

One can show that f is regular at x if there is an affine neighborhood U ⊂ Amof x in X and an affine neighborhood V ⊂ An of φ(x) in Y such that φ sends U

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into V and such that φ can be defined on U by n polynomials in m variables.

Definition 1.0.23. A rational map from a variety X to a variety Y is a map thatis a morphism on some nonempty open subset of X . A rational map φ : X →Y is said to be dominant if φ(U) is dense in Y for some (and consequentlyevery) nonempty open set U ⊂ X on which it is a morphism. A birationalmap is a rational map that has a rational inverse. Two varieties are said to bebirationally equivalent if there is a birational map between them. For φ : X →Y a rational map, the largest open subset U on which φ is a morphism, is calledthe domain of φ.Theorem 1.0.24. i) Let V be an affine variety. Then O(V ) ∼= k[V ].

ii) Let V,W be affine varieties. The natural map

Mor(V,W )→ Homk−Alg(k[W ], k[V ]),φ 7→ (f 7→ f ◦ φ),

is a bijection. The association V → k[V ] is a contravarient functor that in-duces an equivalence between the category of affine varieties and the categoryof finitely generated integral k-algebras.

Thus an affine variety is completely determined by its ring of regular functions.

Lemma 1.0.25. A regular function on a projective variety is constant.Theorem 1.0.26. The image of a projective variety by a morphism is a projectivevariety. More generally, if X is a projective variety, the projection X × Y → Yis a closed map.

Recall that if p is a prime ideal in a ring A, then

Ap = {a/b : a, b ∈ A, b 6∈ p}If p is a homogeneous ideal in a graded ring A, the homogeneous localized ringat p is

A(p) = {a/b : a, b ∈ A, deg(a) = deg(b), b 6∈ p}.In both cases, the local ring is a subring of Frac(A).

Theorem 1.0.27. i) Let P be a point on an affine variety V , and letMP be theideal of functions in k[V ] that vanish at P . Then

OP,V = k[V ]MP and k(V ) = Frac(k[V ]).ii) Let P be a point on a projective variety V , and letMP be the ideal generatedby homogeneous polynomials vanishing at P . Then

OP,V = S[V ](MP ) and k(V ) = S(V )((0)).

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Theorem 1.0.28. Let f : V → W be a rational map between two varieties.i) If f is regular at P and Q = f(P ), then the map

f ∗ : OQ,W → OP,V , f ∗ : g 7→ g ◦ f,

is a homomorphism of local rings. In particular, f ∗(MQ) ⊂MP .ii) If f is dominant, then f ∗ defines a field homomorphism k(W ) ↪→ k(V ).

Conversely, every such field homo corresponds to a dominant rational map. I.e.,the association X → k(X) is a contravariant functor that induces an equiv-alence between the category of varieties with dominant rational maps and thecategory of fields of finite transcendence degree over k.

iii) In particular, two varieties are birationally equivalent if and only if theirfunction fields are isomorphic.

Definition 1.0.29. Let φ : V → W be a morphism of affine varieties, and usethe map φ∗ : k[W ] → k[V ] (from earlier) to make k[V ] into a k[W ]-module.The morphism φ is called finite if k[V ] is a finitely generated k[W ]-module. Amorphism φ : V → W between varieties is finite if for every affine open subsetU ⊂ W , the set φ−1(U) is affine and the map φ : φ−1(U)→ U is finite.

Notice that a map φ between affine varieties is dominant iff φ∗ is injective sowe say that φ is finite surjective if it is finite and φ∗ is injective. There is aninteger d and a nonempty open U ⊂ φ(V ) such that #φ−1(x) = d for all x ∈ U .The degree d can be described algebraically as the degree of the associated fieldextension, and we define this quantity to be the degree of the finite map φ,

deg(φ) = [k(V ) : φ∗k(W )].

Definition 1.0.30. The dimension of a variety V defined over k is the transcen-dence degree of its function field k(V ) over k. The dimension of an algebraic setis the maximum of the dimensions of its irreducible components.

Both An and Pn have dimension n.

Proposition 1.0.31. A variety V of dimension n− 1 is birational to a hypersur-face in An (or Pn).

Definition 1.0.32. The height of a prime ideal p in A is the supremum of all nsuch that there exists a chain of distinct prime ideals p0 ⊂ · · · ⊂ pn = p. TheKrull dimension of the ringA is the supremum of the heights of its prime ideals.

Theorem 1.0.33. i) Let V be an affine algebraic set. Then

dim(V ) = Krull dim(k[V ]).

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ii) Let V be an affine variety and let p be a prime ideal in k[V ]. Then

height(p) +Krull dim(k[V ]/p) = Krull dim(k[V ]).

iii) Let W be a subvariety of V . Then

Krull dim(OW,V ) = dim(V )− dim(W ).

Corollary 1.0.34. Let V be a variety, and let W be a closed algebraic subset ofV . If W 6= V , then dimW < dimV .

Definition 1.0.35. A variety of dimension one is called a curve, and a varietyof dimension two is called a surface. Of course, if we are working over the fieldk = C, then a curve is also sometimes called a Riemann surface.

Proposition 1.0.36. Let V be an affine variety of dimension ` in An, and letZ be a hypersurface in An. Then either V is contained in Z, or else all of thecomponents of V ∩Z have dimension exectly `−1. (Note that V ∩Z may consistof zero components).

Theorem 1.0.37. Let V and W be affine varieties in An of dimensions ` and m,respectively. Then every component of V ∩W has dimension at least `+m−n.

Theorem 1.0.38. Let V and W be projective varieties in Pn of dimension `and m, respectively. Then every component of V ∩ W has dimension at least`+m− n. Furthermore, if `+m− n ≥ 0, then V ∩W is not empty.

Theorem 1.0.39. Let f : X → Y be a surjective morphism of varieties.i) dim(f−1{y}) ≥ dim(X)− dim(Y ) for all y ∈ Y .ii) There is a nonempty open subset U ⊂ Y such that

dim(f−1{y}) = dim(X)− dim(Y ) for all y ∈ U.

Let V be an affine variety defined by the equations

f1(x1, . . . , xn) = · · · = fm(x1, . . . , xn) = 0.

A natural way to define the tangent space to V at the point P = (a1, . . . , an) isby the equations

n∑i=1

∂fj∂xi

(P )(xi − ai) = 0 for 1 ≤ j ≤ m.

The definition of tangent space is independent of the defining equations for V ,the following is valid for arbitrary varieties.

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Definition 1.0.40. Let P be a point on a variety V . The tangent space to V atP is the k-vector space

TP (V ) = Homk(MP,V /M2P,V , k).

I.e., the tangent space is the dual of the vector spaceMP,V /M2P,V . We natruallycall that vector space the cotangent space to V at P. Both spaces are k-vectorspaces, since OP,V /MP,V ∼= k.

Theorem 1.0.41. Let V be a variety. Then dim(TP (V )) ≥ dim(V ) for allP ∈ V . Furthermore, there exists a nonempty open subset U ⊂ V such thatdim(TP (V )) = dim(V ) for all P ∈ U .

Definition 1.0.42. A point P on a variety V is singular if dim(TP (V )) >dim(V ), and it is nonsingular/smooth if dim(TP (V )) = dim(V ). The vari-ety V is called nonsingular/smooth if all of its points are nonsingular.

Lemma 1.0.43. (Jacobian criterion) Let V be an affine variety defined by theequations

f1(x1, . . . , xn) = · · · = fm(x1, . . . , xn) = 0,and let P = (a1, . . . , an) be a point on V . Then P is a smooth point if and onlyif

Rank(∂fj∂xi

(P )

)1≤j≤m,1≤i≤n

= n− dim(V ).

Definition 1.0.44. Consider a rational map f : V → W that is regular at Pand let Q = f(P ). We have seen that f induces a homomorphism of local rings,f ∗ : OQ,W → OP,V and hence it induces a k-linear map

f ∗ :MQ,W/M2Q,W →MP,V /M2P,V ,

The tangent map df(P ) : TP (V ) → TQ(W ) is the transpose of the mapf ∗ :MQ,W/M2Q,W →MP,V /M2P,V .

Theorem 1.0.45. Let V be a variety and let P ∈ V be a smooth point. Then thelocal ring OP,V is a regular local ring.

Theorem 1.0.46. Let φ be a rational map from a smooth variety V to a projectivevariety. Then

codimV (V \dom(φ)) ≥ 2.In other words, a rational map on a smooth variety is defined except possibly ona set of codimension at least 2.

For a variety X , let f ∈ k(X)∗. For any point x ∈ dom(f) we have a tangent

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map df(x) : Tx(X) → Tf(x)(A1) = k, so df(x) is a linear form on Tx(X). Wemay view df as a map that associates to each point x, a linear form on Tx(X)(i.e., a cotangent vector). We call such a map an abstract differential form.

Definition 1.0.47. A regular differential 1-form on a variety X is an abstractdifferential form ω such that for all x ∈ X there is a neighborhood U of x andregular functions fi, gi ∈ O(U) such that ω =

∑fidgi on U . We denote the

set of regular 1-forms on X by Ω1[X]. It is clearly a k-vector space, and is anO(X)-module.

Definition 1.0.48. Let x be a nonsingular point on a variety X of dimension n.Functions t1, . . . , tn ∈ Ox are called local parameters at x if the ti’s are inMxand if they give a basis ofMx/M2x. The functions give local coordinates on Xif t′i = ti − ti(x) give local parameters at all x in X .

Proposition 1.0.49. Let x be a nonsingular point on X . Then there exist lo-cal parameters t1, . . . , tn at x and a neighborhood U of x such that Ω1[U ] =⊕ni=1O(U)dti.

Recall that ∧rV is the space of r-linear skew-symmetric forms on the vectorspace V .

Definition 1.0.50. An abstract r-form ω on a variety X assigns to each x ∈ Xa linear map ω(x) : ∧rTx(X) → k. A regular r-form ω on X is an abstractr-form such that for all x ∈ X there is a neighborhood U containing x andfunctions fi, gi1,...,ir ∈ O(U) such that

ω =∑

gi1,...,irdfi1 ∧ · · · ∧ dfir .

We will let Ωr[U ] denote the space of regular r-forms on U . It is clearly anO(U)-module. If t1, . . . , tn are local coordinates on U , then

Ωr[U ] = ⊕i1


Definition 1.0.53. An abelian variety is a projective variety that is also an al-gebraic group.

Theorem 1.0.54. (Chevalley) LetG be an algebraic group defined over k. Thereexists a maximal connected affine subgroup H of G. This subgroup H is definedover k and is a normal subgroup of G. The quotient of G by H has a naturalstructure as an abelian variety.

A polynomial in one variable is determined up to a scalar by its roots, countedwith multiplicities. A polynomial in several variables is determined, again up toa scalar, by the hypersurfaces counted with multiplicities on which it vanishes.Further, these hypersurfaces with their multiplicities correspond exactly to thedecomposition of the polynomial into irreducible factors. The theory of divisorsis a device that generalizes this idea to arbitrary varieties, where unique factor-ization no longer holds. Throughout most, k will be algebraically closed.

Definition 1.0.55. LetX be an algebraic variety. The group of Weil divisors onX is the free abelian group generated by the closed subvarieties of codimensionone on X . It is denoted by Div(X). I.e., a divisor is a finite formal sum of theform D =

∑nY Y , where the nY ’s are integers and the Y ’s are codimension-

one subvarieties of X . For example, if X is a curve, then the Y ’s are points, ifX is a surface, then the Y ’s are (irreducible) curves, and so on.

Definition 1.0.56. The support of the divisor D =∑nY Y is the union of all

those Y ’s for which the multiplicity nY is nonzero. It is denoted by supp(D).The divisor is said to be effective/positive if every nY ≥ 0.

Recall that if Y is an irreducible divisor on X , then OY,X is the local ring offunctions regular in a neighborhood of some point of Y . In particular, if thevariety X is nonsingular, or more generally if it is nonsingular along Y , thenOY,X is a DVR. We write ordY : OY,X\{0} → Z for the normalized valuationon OY,X and we extend ordY to the fraction field k(X)∗ in the usual way.

Lemma 1.0.57. The order function ordY : k(X)∗ → Z has the following prop-erties:

i) ordY (fg) = ordY (f) + ordY (g) for all f, g ∈ k(X)∗.ii) Fix f ∈ k(X)∗. There are only finitely many Y ’s with ordY (f) 6= 0.iii) Let f ∈ k(X)∗. Then ordY (f) ≥ 0 iff f ∈ OY,X . Similarly, ordY (f) = 0

iff f ∈ O∗Y,X .iv) Assume further that X is projective, and let f ∈ k(X)∗. Then the follow-

ing are equivalent: (a) ordY (f) ≥ 0 for all Y . (b) ordY (f) = 0 for all Y . (c)f ∈ k∗.

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Definition 1.0.58. Let X be a variety and let f ∈ k(X)∗ be a rational functionon X . The divisor of f is the divisor

div(f) =∑Y

ordY (f)Y ∈ Div(X).

A divisor is said to be principal if it is the divisor of a function. Two divisorsD and D′ are said to be linearly equivalent, denoted by D ∼ D′, if theirdifference is a principal divisor. We also sometimes write (f) for the divisor off . The divisor of zeros of f , denoted by (f)0, and the divisor of poles of f ,denoted by (f)∞ are defined by

(f)0 =∑

ordY (f)>0

ordY (f)Y and (f)∞ =∑

ordY (f)


i) The Ui’s are open sets that cover X .ii) The fi’s are nonzero rational functions fi ∈ k(Ui)∗ = k(X)∗.iii) fif−1j ∈ O(Ui ∩ Uj)∗ (i.e., fif−1j has no poles or zeros on Ui ∩ Uj).

Two collections {(Ui, fi) : i ∈ I} and {(Vj, gj) : j ∈ J} are considered to beequivalent (define the same divisor) if fig−1j ∈ O(Ui ∩ Vj)∗ for all i ∈ I andj ∈ J .

The sum of two Cartier divisors is

{(Ui, fi) : i ∈ I}+ {(Vj, gj) : j ∈ J} = {(Ui ∩ Vj, figj) : (i, j) ∈ I × J}.

Definition 1.0.62. With this operation, the Cartier divisors form a group that wedenote by CaDiv(X). The support of a Cartier divisor is the set of zeros andpoles of the fi’s. A Cartier divisor is said to be effective/positive if it can bedefined by a collection {(Ui, fi) : i ∈ I} with every fi ∈ O(Ui). That is, fi hasno poles on Ui.

Definition 1.0.63. Associated to a function f ∈ k(X)∗ is its Cartier divisor,denoted by

div(f) = {(X, f)}.Such a divisor is called a principal Cartier divisor. Two divisors are said tobe linearly equivalent if their difference is a principal divisor. The group ofCartier divisor classes modulo linear equivalence is called the Picard group ofX and is denoted by Pic(X). (Many texts define it has the group of line bundlesor invertible sheaves on X).

Theorem 1.0.64. Let X be a smooth variety. Then the natural maps

CaDiv(X)→ Div(X) and Pic(X)→ Cl(X)

are isomorphisms.

Definition 1.0.65. Let g : X → Y be a morphism of varieties, letD ∈ CaDiv(Y )be a Cartier divisor defined by {(Ui, fi) : i ∈ I}, and assume that g(X) is notcontained in the support of D. Then the Cartier divisor g∗(D) ∈ CaDiv(X) isthe divisor defined by

g∗(D) = {(g−1(Ui), fi ◦ g) : i ∈ I}.

From the definition, g∗(D + E) = g∗(D) + g∗(E) whenever they are defined,and that if g : X → Y and f : Y → Z are two morphisms of varieties, then(f ◦ g)∗ = g∗ ◦ f ∗. It is also clear that

g∗(div(f)) = div(f ◦ g),

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provided that the rational function f ∈ k(Y ) gives a well-defined rational func-tion on g(X).

Proposition 1.0.66. Let f : X → Y be a morphism of varieties. The mapf ∗ : CaDiv(Y ) → CaDiv(X), which is well-defined only for D such thatf(X) 6⊂ supp(D), induces a (well-defined) homomorphism f ∗ : Pic(Y ) →Pic(X).

Definition 1.0.67. Let f : X → Y be a finite map of smooth projective varieties,let Z be an irreducible divisor on X , and let Z ′ = f(Z) be the image of Z underf . Note that the dimension theorem tells us that Z ′ is an irreducible divisor onY . Let sZ be a generator of the maximal ideal ofOZ,X , and similarly let sZ′ be agenerator of the maximal ideal ofOZ′,Y . (That is, sZ and sZ′ are local equationsfor Z and Z ′.) The ramification index of f along Z is defined to be the integer

eZ = eZ(f) = ordZ(sZ′ ◦ f),

where we recall that ordZ : OZ,X → Z is the valuation on OZ,X . Equivalently,sZ′ ◦ f = useZZ for some function u ∈ O∗Z,X . The map f is said to be ramifiedalong Z if eZ(f) ≥ 2.

Proposition 1.0.68. (Hurwitz formula) Let f : X → Y be a finite map betweensmooth projective varieties.

i) The map f is ramified only along a finite number of irreducible divisors.ii) If we assume further either that the characteristic of k is 0 or that the char

of k does not divide any of the ramification indices, then we have the formula

KX ∼ f ∗(KY ) +∑Z

(eZ(f)− 1)Z.

To each divisor D we associate the vector space of rational functions whosepoles are no worse than D:

Definition 1.0.69. Let D be a divisor on a variety X . The vector space L(D) isdefined to be the set of rational functions

L(D) = {f ∈ k(X)∗ : D + div(f) ≥ 0} ∪ {0}.

The dimension of L(D) as a k-vector space is denoted by `(D).

Lemma 1.0.70. Let X be a variety and let D,D′ ∈ Div(X).i) k ⊂ L(D) iff D ≥ 0.ii) If D ≤ D′, then L(D) ⊂ L(D′).iii) If D′ = D + div(g), then the map f 7→ gf gives an isomorphism of k-

vector spaces L(D′) → L(D). In particular, the dimension `(D) depends onlyon the class of D in Pic(X).

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Now, for a while, we don’t assume k is closed, yet it is merely perfect.

Definition 1.0.71. Let X be a variety defined over k. A divisor D is said to bedefined over k if it is invariant under the action of the Galois group Gk.

If the divisor D is defined over k, we consider the k-vector space Lk(D) definedby

Lk(D) = {f ∈ k(X) : D + div(f) ≥ 0}.

Proposition 1.0.72. Let k be a perfect field, let k be an algebraic closure of k,let X be a variety defined over k, and let D be a divisor on X defined over k.

i) The k-vector space Lk(D) = L(D) has a basis of elements in k(X).I.e., there is a natural identification Lk(D) ⊗k k = L(D), and in particular,dimLk(D) = dimL(D) = `(D).

ii) Assume further that X is projective. If there exists a rational functionf ∈ k(X) with D = div(f), then there exists a rational function f ′ ∈ k(X)with D = div(f ′). I.e., the natural map from Pic(X)k to Pic(X) is injective.

Definition 1.0.73. Let X be a variety of dimension n, and let D1, . . . , Dn ∈Div(X) be irreducible divisors with the property that dim(∩iDi) = 0. Chooselocal equations f1, . . . , fn for D1, . . . , Dn in a neighborhood of a point x ∈ X .The (local) intersection index of D1, . . . , Dn at x is

(D1, . . . , Dn)x = dimk(Ox,X/(f1, . . . , fn)).

Further, it is positive iff x ∈ ∩iDi. We define the intersection index (number)of D1, . . . , Dn to be

(D1, . . . , Dn) =∑x∈X

(D1, . . . , Dn)x.

Theorem 1.0.74. Let X and Y be normal projective varieties of dimension n,and let f : X → Y be a finite morphism. Let D1, . . . , Dn ∈ Div(Y ). Then

(f ∗D1, . . . , f∗Dn)X = deg(f) · (D1, . . . , Dn)Y .

Definition 1.0.75. Let X be a projective variety, let i : Z ↪→ X be a subvarietyof dimension r, and let D ∈ Div(X). The degree of Z with respect to D isdefined to be

degD(Z) = (i∗(D), . . . , i∗(D))Z r times.

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Definition 1.0.76. A linear system on a varietyX is a set of effective divisors alllinearly equivalent to a fixed divisor D and parametrized by a linear subvarietyof P(L(D)) ∼= P`(D)−1. The dimension of the linear system is the dimension ofthe linear subvariety. (Some authors say linear series).

Definition 1.0.77. The set of effective divisors linearly equivalent toD is a linearsystem called the complete linear system of D. It is denoted by |D|.Definition 1.0.78. Let L be a linear system of dimension n parametrized by aprojective space P(V ) ⊂ P(L(D)). Select a basis f0, . . . , fn of V ⊂ L(D). Therational map associated to L, denoted by φL, is the map

φL : X → Pn,

x 7→ (f0(x), . . . , fn(x)).Definition 1.0.79. The set of base points of a linear system L is the intersectionof the supports of all divisors in L. We will say that a linear system is base pointfree if this intersection is empty, and we will say that a divisor D is base pointfree if the complete linear system |D| is base point free.Definition 1.0.80. The fixed component of a linear system L is the largestdivisor D0 such that for all D ∈ L, we have D ≥ D0. If D0 = 0, we say that thelinear system has no fixed component.

Theorem 1.0.81. There is a natural bijection between:i) Linear systems L of dimension n without fixed components.ii) Morphisms φ : X → Pn with image not contained in a hyperplane, up to

projective automorphism. (That is, we identify two rational maps φ, φ′ : X →Pn if there is an automorphism α ∈ PGL(n+ 1) such that φ′ = α ◦ φ.)Definition 1.0.82. A linear system L on a projective variety X is very ampleif the associated rational maps φL : X → Pn is an embedding, that is, φL isa morphism that maps X isomorphically onto its image φL(X). A divisor D issaid to be very ample if the complete linear system |D| is very ample. A divisorD is said to be ample if some positive multiple of D is very ample.

Theorem 1.0.83. A linear system L on a variety X is very ample if and only ifit satisfies the following two conditions:

i) (Separation of points) For any pair of points x, y ∈ X there is a divisorD ∈ L such that x ∈ D and y 6∈ D.

ii) (Separation of tangent vectors) For every nonzero tangent vector t ∈Tx(X) there is a divisor D ∈ L such that x ∈ D and t 6∈ Tx(D).Corollary 1.0.84. Let D be a divisor on a curve C.

i) The divisor D is base point free if and only if for all P ∈ C we have`(D − P ) = `(D)− 1.

ii) The divisor D is very ample if and only if for all points P,Q ∈ C we have`(D − P −Q) = `(D)− 2. (Note that we allow P = Q, which corresponds tothe separation of tangent vectors condition).

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Theorem 1.0.85. Every divisor can be written as the difference of two (very)ample divisors. More precisely, letD be an arbitrary divisor and letH be a veryample divisor.

i) There exists an m ≥ 0 such that D +mH is base point free.ii) If D is base point free, then D +H is very ample.

Proposition 1.0.86. i) Let f : X → Y be a morphism between two projectivevarieties. If D is a base point free divisor on Y , then f ∗D is a base point freedivisor on X .

ii) Let f : X → Y be a finite morphism between two projective varieties. IfD is an ample divisor on Y , then f ∗D is an ample divisor on X .

Theorem 1.0.87. (Enriques-Severi-Zariski) LetX ↪→ Pn be a normal projectivevariety. There exists an integer d0 = d0(X) such that for all integers d ≥ d0, thelinear system LX(d) is a complete linear system. I.e., if D is an effective divisoron X that is linearly equivalent to d times a hyperplane section, then there is ahomogeneous polynomial F of degree d such that D = (F )X .

Lemma 1.0.88. Let I be a homogeneous ideal in k[x0, . . . , xn], and let A =k[x0, . . . , xn]/I . There exists an integer d0 = d0(I) such that for all d ≥ d0, allN ≥ 0, and all F ∈ Frac(A),

xN0 F, xN1 F, . . . , x

Nn F ∈ AN+d ⇒ F ∈ Ad.

Corollary 1.0.89. Let D be a divisor on a projective variety. Then `(D) =dimL(D) is finite.

Definition 1.0.90. Let X be a topological space. A presheaf F on X consistsof the following data:

i) For every open subset U in X , a set F(U).ii) For all open subsets V ⊂ U ⊂ X , a map rU,V : F(U)→ F(V ) satisfying

rU,U = idF(U) and rU,W = rV,W ◦ rU,V .

In many cases we may think of the maps rU,V as restriction maps. If the F(U)’shave some additional structure, for example if they are groups, rings, etc, thenwe speak of a presheaf of groups, rings etc.

Definition 1.0.91. A morphism of presheaves f : F1 → F2 is a collection ofmaps f(U) : F1(U) → F2(U) such that for every V ⊂ U , the maps f(U) andf(V ) are compatible with restrictions r2U,V ◦ f(U) = f(V ) ◦ r1U,V . If the Fi’sare presheaves of groups (resp rings, modules), then we insist that the f(U)’sshould be group (resp ring, module) homomorphisms.

Definition 1.0.92. Let X be a topological space. A sheaf F on X is a presheafwith the property that for every open subset U ⊂ X and every open coveringU = ∪iUi, the following two properties are true:

i) Let x, y be elements of F(U) such that rU,Ui(x) = rU,Ui(y) for all i. Thenx = y.

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ii) Let xi ∈ F(Ui) be a collection of elements such that for every pair ofindices i, j, we have rUi,Ui∩Uj(xi) = rUj ,Ui∩Uj(xj). Then there exists a (unique)x ∈ F(U) such that rU,Ui(x) = xi for all i.

The fundamental example in classical algebraic geometry is the sheaf of regularfunctions on a variety X equipped with the Zariski topology. Thus OX is thesheaf defined by

OX(U) = {regular functions on U}, and rU,V is the restriction function from U to V . This construction is so funda-mental that from the point of view of schemes, a variety is a pair (X,OX). Thesheaf of invertible functions O∗X associates to an open set U the set of regularfunctions without zeros on U . Notice that O∗X(U) is exactly the group of unitsin the ring OX(U). On a variety X , the sheaf of rational functions KX attachesto each open set U the set of rational functions on U . It is a constant sheaf inthe sense that all of the maps rU,V are isomorphisms. There is an obvious way toform direct sum and tensor products of two sheaves of modules:

(F ⊕ G)(U) = F(U)⊕ G(U) and (F ⊗ G)(U) = F(U)⊗ G(U).

Definition 1.0.93. The stalk of a sheaf F at a point x ∈ X , denoted by Fx, isthe direct limit of the F(U)’s over all open sets U containing x. Thus

Fx = lim−−→x∈U

F(U),

where the limit is taken with respect to the restriction maps rU,V . Intuitively,an element of the stalk, is an element s ∈ F(U) for some open set containingx, where we identify s and s′ ∈ F(U ′) if s and s′ have the same restriction toU ∩ U ′. It is clear that the stalk of a sheaf of groups (resp. rings, modules) is agroup (resp. ring, module). The elements of Fx are called germs at x. If x ∈ U ,we ge a map F(U)→ Fx. The image of s ∈ F(U) in Fx is called the germ of sat x.

As an example, The stalk of OX at x is just the local ring Ox,X .

Definition 1.0.94. Let F be a sheaf on X . The set of global sections of F is theset F(X). This set is also frequently denoted by Γ(X,F).

For example, if X is an affine variety with coordinate ring R = k[X], thenΓ(X,OX) = R and Γ(X,O∗X) = R∗. However, if X is a projective variety, thenΓ(X,OX) = k and Γ(X,O∗X) = k∗.

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Definition 1.0.95. Let X be a variety. An OX-module is a sheaf F on X suchthat

i) For every U ⊂ X,F(U) is a module over the ring OX(U).ii) For every V ⊂ U ⊂ X , the map rU,V : F(U) → F(V ) is a homomor-

phism of modules. I.e., if s1, s2 ∈ F(U) and f1, f2 ∈ OX(U), thenrU,V (f1s1 + f2s2) = rU,V (f1)rU,V (s1) + rU,V (f2)rU,V (s2).

For example, the sheaves KX and ΩrX are clearly OX-modules. Similarly, thedirect sum OX ⊕ · · · ⊕ OX = OrX is an OX-module called a free OX-moduleof rank r.

Definition 1.0.96. Let F be an OX-module on X . We say that F is locally freeif each point inX has a neighborhood over whichF is free. The rank of a locallyfree sheaf F is the integer r such that F(U) ∼= OX(U)r for all sufficiently smallopen sets U . A locally free sheaf of rank 1 is called an invertible sheaf (orsometimes line sheaf).

The reason that locally free sheaves of rank 1 are called ’invertible’ is becausethey are the sheavesF for which there exists another sheafF’ such thatF⊗F ′ ∼=OX . Thus the set of invertible sheaves naturally form a group, using tensor prod-uct as the group law and OX as the identity.

We now reinterpret the notion of a Cartier divisor by associating an invertiblesheaf to each Cartier divisor. Let D = {(Ui, fi) : i ∈ I} be a Cartier divisor. Wedefine the sheaf LD to be the subsheaf of KX determined by the conditions

LD(Ui) =1

fiOX(Ui) for all i ∈ I.

The association Cl(D) 7→ LD defines an isomorphism from Pic(X) to thegroup of invertible sheaves.

Definition 1.0.97. A vector bundle of rank r over a variety X is a variety Eand a morphism p : E → X with the following properties:

i) Each fiber Ex = p−1{x} is a vector space of dimension r.ii) The fibration p is locally trivial. This means that for each point x ∈ X

there is a neighborhood U containing x over which the fibration is trivial. Avector bundle of rank 1 is called a line bundle.Definition 1.0.98. Let p : E → X and p′ : E ′ → X ′ be vector bundles. Amorphism of vector bundles is a pair of morphisms f : E → E ′ and f : X →X ′ such that f ◦p = p′◦f and such that for every x ∈ X , the map fx : Ex → E ′xis a linear transformation of vector spaces. The trivial bundle of rank r over Xis X × Ar → X .

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Definition 1.0.99. Let p : E → X be a vector bundle. A section of E is amorphism s : X → E such that p ◦ s = idX . Similarly, a rational section of Eis a rational map s : X → E such that p ◦ s = idX .

Let D be a Cartier divisor. The line bundle associated to D will be denoted byO(D).

Theorem 1.0.100. The association D 7→ O(D) induces a functorial isomor-phism between the group of Cartier divisor classes and the group of ismorphismclasses of line bundles on X . More precisely,

O(D +D′) = O(D)⊗O(D′) and O(−D) = O(D)∨.

Further, O(f ∗D) = f ∗O(D) for any morphism f of varieties.

We see that the space of sections Γ(X,O(D)) is in bijection with the functionsin L(D). A linear system on X is thus given by choosing a vector subspace ofΓ(X,E) for some line bundle E on X .

It is classical to denote by OPn(1) or O(1) the line bundle associated to ahyperplane. We letO(d) denote the line bundle obtained by tensoringO(1) withitself d times. The global sections of O(d) are the homogeneous polynomials ofdegree d,

Γ(Pn,O(d)) = ⊕i1+···+in=dkX i11 · · ·X inn .

Corollary 1.0.101. Let X ↪→ Pn be a normal projective variety and let Dbe a hyperplane section. Then for all sufficiently large d, the restriction mapΓ(Pn,O(d)) → Γ(X,O(dD)) is surjective. I.e., every section (of a suitablepower) of O(D) is given by homogeneous polynomials.

Definition 1.0.102. A curve C is a variety of dimension one, so its function fieldk(C) is of transcendence degree one. It follows that k(C) is algebraic over anysubfield k(x) generated by a nonconstant function x ∈ k(C). Hence we maywrite k(C) = k(x, y), where x and y are nonconstant functions on C satisfyingan algebraic relation P (x, y) = 0. Let C0 ⊂ A2 denote the affine plane curvedefined by P , and let C1 ⊂ P2 be the projective plane curve defined by thehomogenized polynomial Zdeg PP (X/Z, Y/Z). Clearly, C is birational to bothC0 and C1. Any curve birational to C is called a model of C, so we can say thatevery curve has a plane affine model and a plane projective model.

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Definition 1.0.103. An ordinary singularity is a singularity whose tangent coneis composed of distinct lines. The multiplicity of an ordinary singularity is thenumber of lines in its tangent cone.

Definition 1.0.104. A Cremona/quadratic transformation is a birational in-volution from P2 to P2 that, after a linear change of variables on the domain andrange, is defined by Q(X, Y, Z) = (Y Z,XZ,XY ).

Theorem 1.0.105. An algebraic curve is birational to a plane projective curvewith only ordinary singularities. More precisely, any plane curve can be trans-formed by a finite sequence of Cremona transformations into a plane curve withonly ordinary singularities.

Theorem 1.0.106. A rational map from a smooth curve to a projective varietyextends to a morphism defined on the whole curve.

Corollary 1.0.107. A birational morphism between two smooth projective curvesis an isomorphism.

Theorem 1.0.108. Any algebraic curve is birational to a unique (up to isomor-phism) smooth projective curve.

Definition 1.0.109. A divisor on a smooth projective curve C is simply a finiteformal sum D =

∑nPP , and we can define the degree of D to be deg(D) =∑

nP . We denote a canonical divisor on C by KC .

Recall that L(D) = {f ∈ k(C) : (f)+D ≥ 0} is a vector space of finite dimien-sion `(D). The Riemann-Roch theorem allows us to compute this dimension.

Theorem 1.0.110. (Riemann-Roch theorem) LetC be a smooth projective curve.There exists an integer g ≥ 0 such that for all divisors D ∈ Div(C),

`(D)− `(KC −D) = deg(D)− g + 1.

Definition 1.0.111. The integer g is called the genus of the smooth projectivecurve C. When C is not necessarily smooth or projective, its genus is defined tobe the genus of the smooth projective curve that is birational to C.

Corollary 1.0.112. Let C be a smooth projective curve of genus g. Then

`(KC) = g and deg(KC) = 2g − 2.

Corollary 1.0.113. Let C be a smooth projective curve of genus g and let D ∈Div(C).

i) If deg(D) < 0, then `(D) = 0.ii) If deg(D) ≥ 2g − 1, then `(D) = deg(D)− g + 1.iii) (Clifford’s theorem) If `(D) 6= 0 and `(KC − D) 6= 0, then we have

`(D) ≤ 12

deg(D) + 1.

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Corollary 1.0.114. Let C be a smooth projective curve of genus g and let D ∈Div(C).

i) If deg(D) ≥ 2g, then D is base point free.ii) If deg(D) ≥ 2g + 1, then D is very ample.iii) D is ample if and only if deg(D) > 0.

Theorem 1.0.115. (Riemann-Hurwitz formula) Let C be a curve of genus g, letC ′ be a curve of genus g′, and let f : C → C ′ be a finite separable map of degreed ≥ 1. For each point P ∈ C, write eP for the ramification index of f at P , andassume either that char(k) = 0 or else that char(k) does not divide any of theeP ’s. Then

2g − 2 = d(2g′ − 2) +∑P∈C

(eP − 1).

Theorem 1.0.116. Let C be a smooth projective plane curve of degree n. Thenthe genus g of C is given by the formula

g =(n− 1)(n− 2)

2.

Theorem 1.0.117. Let C be a projective plane curve of degree n with only ordi-nary singularities. Then its genus is given by the formula

g =(n− 1)(n− 2)

2−∑P∈S

mP (mP − 1)2

where S is the set of singular points and mP the multiplicity of C at P .

Theorem 1.0.118. Let C be a smooth projective curve of genus 0 defined over afield k.

i) The curve C is isomorphic over k to a conic in P2.ii) The curve C is isomorphic over k to P1 if and only if it possesses a k-

rational point.

Notice in particular that over an algebraically closed field, all curves of genus 0are isomorphic to P1.

Definition 1.0.119. A curve is said to be rational if it is birational to the pro-jective line.

Theorem 1.0.120. (Hasse principle) A conic defined over a number field k hasa k-rational point if and only if it has a rational point over all completions of k.

Lemma 1.0.121. Let C be a smooth projective curve. Then the following areequivalent.

i) C has genus 0.ii) There exists a point P ∈ C such that `(P ) = 2.iii) For every point P ∈ C we have `(P ) = 2.

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Theorem 1.0.122. Let C be a curve of genus 1 defined over a field k, and letP0 ∈ C(k). Then there exist constants a1, a2, a3, a4, a6 ∈ k such that C isisomorphic over k to the smooth plane cubic given by the equation

E : Y 2Z + a1XY Z + a3Y Z2 = X3 + a2X

2Z + a4XZ2 + a6Z

3

Under this isomorphism, the point P0 is mapped to the inflection point (X, Y, Z) =(0, 1, 0) ∈ E.

If the characteristic of k is not 2 or 3, then by completing the square in y andthe cube in x, one can find for E a curve given by an equation of the form

E : Y 2Z = X3 + AXZ2 +BZ3, A,B ∈ k.For a curve in this form, the nonsingularity of E is equivalent to the nonvanish-ing of the discriminant 4A3 + 27B2 6= 0.Definition 1.0.123. An elliptic curve is a pair (C,P0), where C is a (smoothprojective) curve of genus 1 and P0 is a point on C. The elliptic curve is definedover k if the curve C is defined over k and also P0 ∈ C(k). Thus the previoustheorem says that every elliptic curve is isomorphic to a smooth plane cubic withP0 corresponding to an inflection point ’at infinity’. An equation of the form

Y 2Z + a1XY Z + a3Y Z2 = X3 + a2X

2Z + a4XZ2 + a6Z

3

orY 2Z = X3 + AXZ2 +BZ3

is called a Weierstrass equation for E. Frequently, these equations are writtenin affine coordinates (i.e., by setting Z = 1) where it is understood that there isone additional point P0 = (0, 1, 0) at infinity.

Assuming that the characteristic of k is not 2 or 3, we will work with an ellipticcurve given by the affine Weierstrass equation

E : y2 = x3 + Ax+B.

We begin by defining an involution [−1] on E,[−1] : E → E, (x, y) 7→ (x,−y).

Next we define a tangent and chord operation, which we will denote by +. LetP,Q ∈ E. If P and Q are distinct, let L be the line through P and Q, while ifP = Q, let L be the tangent line to E at P . The line L will meet E at a thirdpoint R, counting multiplicities, and then we set

P +Q = [−1]R.With these operations, the points of E become a group. The group law is alge-braic and intrinsic, i.e., it is given by rational functions and does not depend onthe equation or embedding of E. The group law depends only on the abstractcurve E and the choice of the point P0.

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Theorem 1.0.124. Let E be a smooth projective cubic given by a Weierstrassequation

Y 2Z = X3 + AXZ2 +BZ3.

Then the maps (P,Q) → P + Q and P → [−1]P defined above give E thestructure of a commutative algebraic group with identity element P0 = (0, 1, 0).Furthermore, the map

κ : E → Pic0(E), P 7→ divisor class of (P )− (P0),

is a group isomorphism.

Definition 1.0.125. A curve of genus g ≥ 2 is called a hyperelliptic curve if itis a double covering of the projective line.

Theorem 1.0.126. Let C be a smooth projective curve of genus g.i) The canonical divisor KC is base point free if and only if g ≥ 1.ii) The canonical divisor KC is ample if and only if g ≥ 2.iii) The canonical divisor KC is very ample if and only if g ≥ 3 and the curve

is not hyperelliptic.iv) The bicanonical divisor 2KC is very ample if and only if g ≥ 3.v) The divisor 3KC is very ample if and only if g ≥ 2.

Theorem 1.0.127. (Bezout’s theorem) Let C and D be curves on the surface P2defined by irreducible equations of degree m and n. Then the intersection indexof C and D is C ·D = mn. In particular, if C 6= D, then the number of pointsof intersection counted with multiplicities is mn.

Theorem 1.0.128. (Adjunction formula) Let S be a smooth projective surface,let KS be a canonical divisor on S, and let C be a smooth irreducible curve ofgenus g on S. Then

C2 + C ·KS = 2g − 2.Theorem 1.0.129. (Riemann-Roch for surfaces) Let S be a smooth surface, andlet KS be a canonical divisor on S. There exists an integer pa(S) such that forany divisor D ∈ Div(S),

`(D)− s(D) + `(KS −D) =1

2D · (D −KS) + 1 + pa(S)

for some nonnegative integer s(D).

Proposition 1.0.130. Let C be a smooth projective curve of genus g, fix a pointP0 on C, and let S = C × C. Define divisors D1, D2,∆ ∈ Div(S) by

D1 = C × {P0}, D2 = {P0} × C, ∆ = {(P, P ) ∈ S : P ∈ C}.

Notice that ∆ is the diagonal of C × C.i) D21 = D

22 = 0.

ii) D1 ·D2 = ∆ ·D1 = ∆ ·D2 = 1.iii) ∆2 = 2− 2g.

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Definition 1.0.131. LetG1 andG2 be two algebraic (or analytic) groups. A mapα ∈ Hom(G1, G2) is called an isogeny if it is surjective, has finite kernel, anddimG1 = dimG2. The cardinality of ker(α) is called the degree of α.

Definition 1.0.132. An entire function f on Cg is a theta function relative tothe lattice Λ if it satisfies a functional equation of the form

f(z + λ) = exp(gλ(z))f(z) for all λ ∈ Λ,

where gλ is an affine function on z. That is, g : Cg → C has the property thatg(z+w) +g(0) = g(z) +g(w) for all z, w ∈ Cg. The function exp is sometimescalled the automorphy factor of the theta function. A theta function of the formexp(Q(z) + R(z) + S), where Q is a quadratic form, R is a linear form, and Sis a constant, will be called a trivial theta function.

Let θ be a theta function with divisorD for a lattice Λ in V = Cg. We write L(θ)for the vector space of all theta functions with the same functorial equation. It isfinite dimensional. Choosing a basis θ1, . . . , θn we get a holomorphic map

φD : V/Λ→ Pn(C), z mod Λ 7→ (θ0(z), . . . , θn(z)).

Definition 1.0.133. The divisor D on the torus V/Λ is very ample if the mapφD described above is an embedding. The divisor D is ample if some positivemultiple of D is very ample.

Lemma 1.0.134. (Frobenius) Let Λ be a free abelian group of rank 2g (i.e.,Λ ∼= Z2g). Let E be a nondegenerate bilinear alternating form on Λ with val-ues in Z. There exist positive integers d1, . . . , dg with di | di+1 and a basise1, . . . , eg, f1, . . . , fg of Λ such that

E(ei, ej) = E(fi, fj) = 0 and E(ei, fj) = di, if i = j, otherwise 0.

The product d1 · · · dg is the square root of the determinant of E.

Definition 1.0.135. Let E be a nondegenerate bilinear alternating form on Λ ∼=Z2g with values in Z. A basis (as in the lemma) is called a Frobenius basis, andthe di’s are called the invariants of E. Further, we define the Pfaffian of E tobe the quantity

Pf(E) = d1d2 · · · dg =√

det(D).

Theorem 1.0.136. (Riemann-Roch for abelian varieties) Let D be a divisor onan abelian variety, let HD be the nondegenerate Riemann form for D, let ED =ImHD, and let Pf(ED) be its Pfaffian. Then `(D) = Pf(ED).

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Theorem 1.0.137. (Lefschetz) Let θ be a theta function with divisor D and non-degenerate Riemann form H .

i) The divisor 2D is base point free.ii) The divisor 3D is very ample.

Corollary 1.0.138. Let φ : A → B be a morphism between two abelian vari-eties. Then φ is the composition of a tranlation and a homomorphism.

Lemma 1.0.139. An abelian variety is a commutative algebraic group.

Lemma 1.0.140. (Weil) A rational map from a smooth variety into an algebraicgroup either extends to a morphism or is undefined on a set of pure codimensionone.

Corollary 1.0.141. A rational map from a smooth variety into an abelian varietyextends to a morphism.

Proposition 1.0.142. Let A be an abelian variety, and let f : A → Y be amorphism. Then there is an abelian subvariety B of A such that for x ∈ A, theconnected component of f−1{f(x)} containing x is equal to B + x.

Since abelian varieties are smooth, we don’t need to worry about distinguishingbetween Cartier and Weil divisors, so we will write Pic(A) for the divisor classgroup of A and we will use ∼ to denote linear equivalence. We first definevarious projection summation maps. Thus for any subset I of {1, 2, 3}, we definea map

sI : A× A× A→ A, sI(x1, x2, x3) =∑i∈I

xi.

For example, s13(x1, x2, x3) = x1 + x3 and s2(x1, x2, x3) = x2.

Theorem 1.0.143. (Theorem of the cube on abelian varieties) Let A be anabelian variety. Then for every divisor D ∈ Div(A), the following divisor classrelation holds in A× A× A:

s∗123D − s∗12D − s∗13D − s∗23D + s∗1D + s∗2D + s∗3D ∼ 0.

Theorem 1.0.144. (Theorem of the cube) Let X, Y , and Z be projective vari-eties, and let (x0, y0, z0) ∈ X × Y × Z. Let D be a divisor on X × Y × Zwhose linear equivalence class becomes trivial when restricted to each of thethree slices

X × Y × {z0}, X × {y0} × Z, and {x0} × Y × Z.

Then D is linearly equivalent to zero on X × Y × Z.

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Corollary 1.0.145. Let A be an abelian variety, let V be an arbitrary variety,and let f, g, h : V → A be three morphisms from V to A. Then for any divisorD ∈ Div(A),

(f +g+h)∗D− (f +g)∗D− (f +h)∗D− (h+g)∗D+f ∗D+g∗D+h∗D ∼ 0.

Corollary 1.0.146. (Mumford’s formula) Let D be a divisor on an abelian vari-ety A, and let [n] : A→ A be the multiplication-by-n map. Then

[n]∗(D) ∼(n2 + n

2

)D +

(n2 − n

2

)[−1]∗(D).

In particular,

[n]∗(D) ∼

{n2D if D is symmetric ([−1]∗D ∼ D)nD if D is antisymmetric ([−1]∗D ∼ −D)

Theorem 1.0.147. Let A be an abelian variety of dimension g over an alge-braically closed field k of characteristic p ≥ 0.

i) The multiplication-by-n map [n] : A→ A is a degree n2g isogeny.ii) Assume either that p = 0 or that p - n. Then

A[n] = ker[n] ∼= (Z/nZ)2g.

iii) If p > 0, then A[pt] ∼= (Z/ptZ)r for some integer 0 ≤ r ≤ g.Theorem 1.0.148. (Theorem of the square) Let A be an abelian variety, and foreach a ∈ A, let ta : A→ A be the translation-by-a map ta(x) = x+ a. Then

t∗a+b(D) +D ∼ t∗a(D) + t∗b(D) for all D ∈ Div(A) and a, b ∈ A.

In other words, for any divisor class c ∈ Pic(A), the map

Φc : A→ Pic(A), a 7→ t∗a(c)− c,

is a group homomorphism.

For any divisor D ∈ Div(A), we let

ΦD : A→ Pic(A), a 7→ class (t∗a(D)−D),

be the homomorphism from earlier, and we let K(D) = ker(ΦD). The groupK(D) can be used to give an ampleness criterion for divisors on abelian vari-eties.

Theorem 1.0.149. Let D be an effective divisor on an abelian variety A. Thenthe linear system |2D| is base-point free, and the following are equivalent:

i) D is ample.ii) The group K(D) = {a ∈ A : t∗a(D) ∼ D} is finite.iii) The stabilizer G(D) = {a ∈ A : t∗a(D) = D} is finite.iv) The morphism A→ PL(2D) associated to 2D is a finite morphism.

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Definition 1.0.150. LetA be an abelian variety. The group Pic0(A) is the groupof translation-invariant divisor classes,

Pic0(A) = {c ∈ Pic(A) : t∗ac = c for all a ∈ A}.

Theorem 1.0.151. Let A be an abelian variety, let c ∈ Pic(A), and let Φc bethe earlier homomorphism.

i) The image of Φc lies in Pic0(A).ii) If nc ∈ Pic0(A) for some integer n 6= 0, then c ∈ Pic0(A).iii) If the divisor class c is ample, then Φc : A → Pic0(A) is surjective and

has a finite kernel.

The Neron-Severi group ofA denoted byNS(A) is the quotient groupNS(A) =Pic(A)/P ic0(A).

Definition 1.0.152. An abelain variety Â is called the dual abelian variety ofA if there exists a divisor class P on A× Â such that the maps

Â→ Pic0(A), â 7→ i∗â(P),

andA→ Pic0(Â), a 7→ i∗a(P),

are both bijections. (Here iâ : A → A × Â is the map iâ(a) = (a, â), andia : Â → A × Â is the map ia(â) = (a, â).) The divisor class P is called thePoincare divisor class.

Theorem 1.0.153. The dual abelian variety Â exists and together with the Poincareclass P ∈ Pic(A× Â) is unique up to isomorphism. Further, the Poincare classP is even.

Proposition 1.0.154. Let π : V → Alb(V ) be the universal map from V to itsAlbenese variety. Then the pullback map π∗ : Pic0(Alb(V )) → Pic0(V ) is anisomorphism. In particular, the Albenese and Picard varieties are dual to eachother.

Theorem 1.0.155. Let C be a smooth projective curve of genus g ≥ 1. Thereexists an abelian variety Jac(C), called the Jacobian of C, and an injectionj : C ↪→ Jac(C), called the Jacobian embedding of C, with the followingproperties:

i) Extend j linearly to divisors on C. Then j induces a group isomorphismbetween Pic0(C) and Jac(C).

ii) For each r ≥ 0, define a subvariety Wr ⊂ Jac(C) by

Wr = j(C) + · · ·+ j(C) r copies.

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(By convention, W0 = {0}.) Then

dim(Wr) = min(r, g) and Wg = Jac(C).

In particular, dim(Jac(C)) = g.iii) Let Θ = Wg−1. Then Φ is an irreducible ample divisor on Jac(C).

Definition 1.0.156. LetR be a commutative ring. The spectrum ofR, Spec(R),is a pair consisting of a topological space (by abuse of notation, also denotedby Spec(R)) and a sheaf O. The topological space Spec(R) is the set of primeideals of R endowed with a topology whose closed sets are the sets V (I) ={p ∈ Spec(R) : I ⊂ p} for any ideal I of R. The sheaf O is characterizedby O(Spec(R)\V ((f))) = Rf for any element f ∈ R, taken with the obviousrestriciton maps.

Proposition 1.0.157. i) The sheafO = OR is entirely characterized by its valueson the principal open subsets Uf = Spec(R)\V ((f)). In fact, one has

O(U) = lim−−−→Uf⊂U

O(Uf ).

ii) For p ∈ Spec(R) the stalk of the sheaf O at p is (isomorphic to) the localring Rp.

Definition 1.0.158. i) A ringed space is a pair (X,OX) consisting of a topo-logical space X and a sheaf of rings OX on X . It is a locally ringed space iffor all x ∈ X , the stalk Ox is a local ring. The sheaf OX is called the structuresheaf of the ringed space.

ii) A morphism of ringed spaces is a pair f, f# : (X,OX) → (Y,OY ),where f : X → Y is continuous and f# : OY → f∗OX is a morphism ofsheaves over Y , i.e., a collection of maps f#(U) : OY (U) → OX(f−1(U))such that rU,V ◦ f#(U) = f#(V ) ◦ rU,V . It is a morphism of locally ringedspaces if further for all x in X , the map f# induces a local ring homomorphismf#x : Of(x) → Ox (i.e., the inverse image of the maximal ideal is the maximalideal).

Definition 1.0.159. A locally ringed space of the form (Spec(R),OR) is calledan affine scheme, where R may be any ring.

Morphisms between affine schemes are described completely analogously tomorphisms between affine varieties. A ring homomorphism φ : R → S in-duces a morphism of locally ringed spaces φsch = (f, f#) : (Spec(S),OS) →(Spec(R),OR) as follows:

If p is a prime ideal of B, set f(p) = φ−1(p).If Ug = Spec(R)\Z(g), then f−1(Ug) = Spec(S)\Z(φ(g)), and we set

f#(Ug) : Rg → Sφ(g) to be the natural map induced by φ on the local rings.

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Proposition 1.0.160. Any morphism of affine schemes Spec(S)→ Spec(R) hasthe form φsch for some ring homomorphism φ : R→ S.

Morphisms between X and Y correspond bijectively to k-morphisms from Xsch

to Y sch since they are both in natural bijection with the k-algebra homomor-phisms from k[Y ] to k[X].

Definition 1.0.161. A scheme is a locally ringed space (X,OX) that can be cov-ered by open subsets U such that (U,OX|U ) is isomorphic to some affine scheme(Spec(R),OR). A morphism of schemes is a morphism of locally ringed spacesthat are schemes. A scheme is called reduced if the rings of the structure sheafcontain no nilpotent elements, irreducible if the associated topological space isirreducible, and integral if it is both reduced and integral.

Definition 1.0.162. Schemes over S, or S-schemes are schemes X that comeequipped with a morphismX → S. In this context, if f : X → S and g : Y → Sare two S-schemes, then an S-morphism is a morphism φ : X → Y satisfyingf = g ◦ φ. This generalizes the notion of varieties and morphisms defined overk, which corresonds to the case S = Spec(k). We also note that every schemeis a Spec(Z)-scheme, because every ring R admits a (unique) homomorphismZ→ R.Definition 1.0.163. To any affine variety X over an algebraically closed field kwe can associate a k-scheme, denoted byXsch, which is simply Spec(k[X]). Theclosed points of Xsch (i.e., the maximal ideals of k[X]) corresond to the pointsof the variety X and are called geometric points. However, Xsch has manyother (nonclosed) points, in fact, one for each irreducible closed subvariety ofX . Of particular interest is the ideal (0), which is dense in Xsch and is calledthe generic point of X .

Schemes are more general than varieties. If k is a field, the scheme Spec(k) hasonly one point. But there are other rings with only one prime ideal, for exampleZ/pnZ and k[X]/(Xn). For ex, X = Spec(Z/pnZ) has only one point, but itis not a variety. It is irreducible, but not reduced when n ≥ 2. The schemeSpec(Z(p)) consists of two points, the generic point η corresonding to the ideal(0) and a unique close point p corresonding to the ideal pZ(p). The schemeSpec(Z) has one generic point η corresponding to the ideal {0}, and all of itsother points are closed and correspond to the prime numbers. The structure sheafis easy to describe:

O(Spec(Z)\{p1Z, . . . , pkZ}) = Z[

1

p1, . . . ,

1

pk

].

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The function field of Spec(Z) (i.e., the stalk at η) is Q. Every scheme is a schemeover Spec(Z).

Definition 1.0.164. The dimension of an irreducible scheme X is the maximallength n of a chain of distinct irreducible closed subsets X0 ⊂ X1 ⊂ · · · ⊂Xn = X . The dimension of a scheme is the maximal dimension of its irreduciblecomponent.

Clearly, dimSpec(R) = Krull dim(R), so the dimension of a variety X is thesame as the dimension of the scheme Xsch. If R is a Dedekind domain, thenSpec(R) is irreducible, reduced, and has dimension 1. In particular, an algebraiccurve and Spec(Z) are two instances of integral schemes of dimension 1. Simi-larly the scheme A1Z = Spec(Z[X]) called the affine line over Z, has dimensiontwo and is analogous to A2k = Spec(k[X, Y ]), the affine plane over the field k.

Field extensions Q ⊂ K and k(C) ⊂ k(C ′) induce finite morphisms Spec(RK)→Spec(Z) and C ′ → C, and the cardinality of the fiber over a closed point is lessthan or equal to [K : Q] or [k(C) : k(C ′)], respectively, with equality at all butfinitely many points. The points where equality fails to hold are called ramifi-cation points.

If X is a variety defined over a field k, a point in X(k) becomes, a morphismSpec(k) → Xsch. It is therefore natural to define a point in X with value inS to be a morphism S → X . I.e., we define X(S) = Mor(S,X). Note thatS can be the spectrum of a ring, or more generally any scheme. The associa-tion S 7→ X(S) defines a contravariant functor from the category of schemesto category of sets. A point x in a scheme defines a local ring Ox, namely thestalk of the structure sheaf at x, hence a maximal idealMx and a residue fieldk(x) = Ox/Mx. In fact, a morphism Spec(K) → X is equiv to the data of apoint x ∈ X and an injection of fields k(x) ↪→ K. For example, one can inter-pret a closed k-point in a varietyX as a Galois conjugacy class of points inX(k).

A varietyX is nonsingular at a pointX if and only if dimX = dimk(Mx/M2x).We define X to be regular at x if dimX = dimk(x)(Mx/M2x).

Definition 1.0.165. Let f : Y → X and g : Z → X be morphisms of schemes.A fibered product of Y and Z over X , denoted by Y ×X Z, is a scheme P withmorphisms p1 : P → Y and p2 : P → Z such that f ◦ p1 = g ◦ p2 and satisfyingthe universal property: For all schemes P ′ with morphisms q1 : P ′ → Y andq2 : P

′ → Z there exists a unique morphism φ : P ′ → P such that q1 = p1 ◦ φand q2 = p2 ◦ φ. Intuitively, at least at the level of closed points, P looks likethe set of pairs (y, z) with f(y) = g(z). Notice that if X, Y, and Z are varieties,

31


then P need not be a variety. However, within the category of schemes, fiberedproducts do exist.

Proposition 1.0.166. Let f : Y → X and g : Z → X be morphisms ofschemes. Then the fibered product Y ×X Z exists and is unique up to canon-ical isomorphism. Further, if X = Spec(R), Y = Spec(A), and Z = Spec(B)are affine, then the fibered product is affine and can be described as Y ×X Z =Spec(A⊗R B).

An important special case of fibered products is extension of scalars. Let Xbe a scheme over a ring R (i.e., X is a Spec(R)-scheme), and let f : R→ R′ bea ring homomorphism. Then f induces a morphism f ∗ : Spec(R′)→ Spec(R),and we extend scalars onX by forming the Spec(R′)-schemeX×Spec(R)Spec(R′).We frequently say that X is an R-scheme, and write the extension as X ×R R′.

Definition 1.0.167. Let f : X → Y be a morphism of schemes and let y bea not necessarily closed point of Y . The point y corresponds to a morphismSpec(k(y)) → Y , and we define the fiber of f over y to be the scheme Xy =X ×Y Spec(k(y)). A family of schemes is just the set of fibers of a morphismof schemes f : X → Y . If Y is irreducible and η is its generic point, we callXη = X ×Y Spec(k(y)) the generic fiber of the family. The fiber Xy over aclosed point y ∈ Y is called the special fiber at y.Proposition 1.0.168. (Zariski’s connectedness principle) Let f : X → S bean irreducible family of projective schemes over an irreducible curve S (i.e., airreducible scheme of dimension 1). Then the generic fiber of f is irreducible.Further, every special fiber of f is connected, and all but finitely many of themare irreducible.

We wish to reverse the above construction by starting with a varietyX and creat-ing a family of schemes whose generic fiber isX . LetK be either a number fieldor the function field k(C) of a smooth projective curve, and let X be a smoothprojective variety defined over K. Let S = Spec(RK) if K is a number field,and let S = Csch if K is a function field. There exists a scheme X → R that isprojective (by which we mean that all fibers are projective varieties) and whosegeneric fiber Xη = X ×S Spec(K) is isomorphic to X .

Definition 1.0.169. Let K and S be as above, and let X be a variety over K. Amodel for X over S is a scheme X → S whose generic fiber is isomorphic toX .

Lemma 1.0.170. Let K and S be as above (i.e., S is regular of dimension 1).Let X be a variety over K, and let X → S be a projective model of X . Then

32


every K-rational point Spec(K) → X of X extends to a morphism (a section)S → X . In other words, there is a natural bijection between X(K) and X (S).Definition 1.0.171. A smooth projective variety X/K has good reduction at xif there exists a projective model of X over Ox whose special fiber is smooth. Ifsuch a model does not exist, we say that X has bad reduction at x.

Proposition 1.0.172. Let X be a smooth projective variety defined over K.i) The variety X has good reduction at all but finitely many points.ii) Let T ⊂ S be the (finite) set of points where X has bad reduction. Let

ST = Spec(RK,T ) in the number field case, and ST = C\T in the function fieldcase. Then there exists a projective model of X over ST all of whose fibers aresmooth.

A one-dimensional affine integral regular scheme is either a smooth curveC overa field k or an open subset of the spectrum of a Dedekind ring, e.g., the ring ofintegers of a number field. If we start with a field K containing an algebraicallyclosed field k, there is a unique (up to isomorphism) smooth projective curveover k having K as its function field. Similarly, if K is a finite extension of Q,the ring of integers RK is the unique maximal order in K, and the associatedscheme is Spec(RK). We call these two situations the geometric case and thearithmetic case respectively.

Definition 1.0.173. An absolute value of a field K is a map | · | : K → R suchthat:

i) |x| ≥ 0 for all x, and |x| = 0 if and only if x = 0.ii) |xy| = |x| · |y|.iii) |x+ y| ≤ |x|+ |y| (triangle inequality)If further we have the stronger inequality

|x+ y| ≤ max(|x|, |y|) for all x, y ∈ K,

then the absolute value is called nonarchimedean. Otherwise, it is calledarchimedean.

Definition 1.0.174. A projective model V → S of V/K is said to be a rela-tively minimal model if it is regular and if every birational morphism from Vto another regular model V ′ is in fact an isomorphism. The model V is saidto be minimal if for any other regular model V ′ there is a birational morphismV ′ → V .Theorem 1.0.175. Let V be a curve of genus g ≥ 1 over K. Then there exists aunique (up to isomorphism) projective minimal model V → S of V .Definition 1.0.176. A curve V defined over a fieldK (number field or 1-dimensionalfunction field) has semistable reduction at p if the special fiber at p of the mini-mal model of V is reduced and has only ordinary double points as singularities.

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Theorem 1.0.177. Let V be a smooth projective curve defined over a numberfield or function field K as above. There exists a finite extension L/K such thatV has semistable reduction at all places L.

Definition 1.0.178. Letting K and S be as above, let V/K be a variety. Ascheme V → S is a Neron model of V/K if it is smooth over S and if forevery smooth scheme X → S with generic fiber X/K and every morphismf : X/K → V/K it is possible to extend f to a morphism of schemes X → V .

Theorem 1.0.179. Let A/K be an abelian variety. Then there exists a Neronmodel A → S of A/K. Furthermore, A is a group scheme over S.

Definition 1.0.180. An abelian varietyA/K has semistable reduction at p if theconnected component of the special fiber Ap of the Neron model is an extensionof an abelian variety by a torus T . It has split semistable reduction if the torusis isomorphic to T = Gsm over the residue field kp.

Theorem 1.0.181. Let A be an abelian variety defined over a number field orfunction field K as above. Then there exists a finite extension L/K such that Ahas (split) semistable reduction at all places of L.

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Chapter 2

Height Functions

35

CHAPTER 2. HEIGHT FUNCTIONS

36

Chapter 3

Rational Points on Abelian Varieties

37

CHAPTER 3. RATIONAL POINTS ON ABELIAN VARIETIES

38

Chapter 4

Diophantine Approx. and IntegralPoints on Curves

39

CHAPTER 4. DIOPHANTINE APPROX. AND INTEGRAL POINTS ON CURVES

40

Chapter 5

Rational Points on Curves of Genusat Least 2

41

CHAPTER 5. RATIONAL POINTS ON CURVES OF GENUS AT LEAST 2

42

Preliminary: Algebraic Geometry ReviewHeight FunctionsRational Points on Abelian VarietiesDiophantine Approx. and Integral Points on CurvesRational Points on Curves of Genus at Least 2

diophantine geometry · 2017. 2. 6. · diophantine geometry travis dirle january 24, 2017. 2....

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