eugenii shustin smoothnessandirreducibilityofvarietiesof

BULLETIN DE LA S. M. F.

EUGENII SHUSTINSmoothness and irreducibility of varieties ofplane curves with nodes and cuspsBulletin de la S. M. F., tome 122, no 2 (1994), p. 235-253<http://www.numdam.org/item?id=BSMF_1994__122_2_235_0>

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Bull. Soc. math. France,

122, 1994, p. 235-253.

SMOOTHNESS AND IRREDUCIBILITY OF VARIETIES

OF PLANE CURVES WITH NODES AND CUSPS

BY

EUGENII SHUSTIN (*)

RESUME. — Soit V(d, 777., k} la variete des courbes projectives planes irreductibles dedegre d n'ayant pour singularites que m nodes et k cusps. Nous montrons que V{d, m, k)est non vide, lisse et irreductible quand m + 2k < ad2 ou a est une constante absolueexplicite. Cette inegalite est optimale quant a Fexposant de d

ABSTRACT. — Let V{d, m, k) be the variety of plane projective irreducible curves ofdegree d with m nodes and k cusps as their only singularities. We prove that V(d, m, k)is non-empty, non-singular and irreducible when m + 2k < ad2, where a is someabsolute explicit constant. This estimate is optimal with respect to the exponent of d

0. IntroductionIn the present article we deal with plane projective algebraic curves

over an algebraically closed field of characteristic 0.It is well-known that the variety of irreducible curves of a given degree

with a given number of nodes is non-singular [9], irreducible [2], andthat each germ of this variety is a transversal intersection of germs ofequisingular strata corresponding to all singular points [9] (from now on,speaking of a variety with the last property, we shall write T- variety, orvariety with property T).

Our goal is a similar result for curves with nodes and ordinary cusps.Let V{d, m, k) denote the set of irreducible curves of degree d with mnodes and k cusps as their only singularities.

(*) Texte recu Ie 29 septembre 1992, revise Ie 3 decembre 1993.E. SHUSTIN, School of Math. Sciences, Tel Aviv University, Ramat Aviv, 69978 TelAviv, Israel. Email : [email protected] : family of singular plane algebraic curves, linear syste, Riemann-Rochtheorem, irreducibility.AMS classification : 14H10.

BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 0037-9484/1994/235/$ 5.00© Societe mathematique de France

236 E. SHUSTIN

• It is known (see [5], [6]) that V(d^m^k) = 0 if

9 5-m+2k > -d2.0 0

• On the other hand (see [13]), V(d,m,k) -^ 0 when

(0.1) m+2k^ ^-d2 +0(d).

Our result is

THEOREM 0.2.• If m + 2k :< aod2^ where

(0.3) ao = 7~v/13 ^ 0.0419,ol

then V(d, m, ̂ ) z5 a non-empty non-singular T-variety of dimension

^d(d+3) -m-2k.

• If m + 2k <, aid2, where

(0.4) ai = ̂ ̂ 0.0089,

^e?2 V(d, m, A;) is irreducible.

Let us make some comments.First, (0.3) implies (0.1), and then V{d,m,k) -^ 0.Let P^, with N = ^d(d +3), be the space of plane curves of degree d.

Let z be a singular point of F e P^. It is well-known (see [2, [9]) that :(1) if z is a node then the germ at F of the variety of curves <I> G P^,

having a node in some neighbourhood of ^, is smooth, has codimension 1,and its tangent space is open in {<!> € P^ | z € ^};

(2) if z is a cusp then the germ at F of the variety of curves ^ G P^,having a cusp in some neighbourhood of z, is smooth, has codimension 2,and its tangent space is open in {<!> C P^ | (^ • ^)(z) > 3} (here andfurther on the notation (F • G) {z) means the intersection number of thecurves F, G at the point z).

Hence the property T implies the smoothness of V(d, m, k) and theexpected value of its dimension given in THEOREM 0.2. Further, it is well-known [15] that V(d, m, k) is a non-singular T-variety, when

(0.5) k < 3d.

TOME 122 — 1994 — ?2

VARIETIES OF PLANE CURVES WITH NODES AND CUSPS 237

Generalizations of this fact to arbitrary singularities, given in [I], [12]are based — in fact — on the same idea. The following conditions aresufficient for the smoothness of V{d^ m, k) and the property T :

m = 0, 2k < (7 ~ vl3)d2 ^ o.0418d2 (see [10], [11]),81

and for the irreducibility of V(d, m, k) :^

m+2k < -d (see [10], [11]),z^

k < 3 (see [7]),

\ (d2 - 4d + 1) < m < \ (d2 - 3d + 2) (see [8]).

The main idea of our proof is as follows. We have to prove the prop-erty T and the irreducibility for V(d, m, k). To any curve F e V(d^ m, k)with nodes z\^..., Zm and cusps W i , . . . , Wk we assign two linear systemsof curves of degree n :

Ai (n, F) = {^ | ̂ i , . . . , Zm C ^, (^ • F){wi) ̂ 3, z = 1 , . . . , A-},

A 2 ( n , F ) = { ^ |2: i , . . . ,^GSing(^) , (^ • F)(w,) > 6, z = l , . . . , A - } .

First we show the non-speciality of Ai (d, P) for any F e ^(d, m, /c), whichmeans according to the Riemann-Roch theorem that

(0.6) dimAi(d.F) = ^d(d + 3) - m - 2k.

On the other hand, Ai(d,.F) is the intersection of the tangent spacesto germs of equisingular strata at F in the space of curves of de-gree d, and (0.6) gives us the transversality of this intersection, or thedesired property T. Then we show that, for any F from some opendense subset U C V(d^m^k\ the system A ' z ( d ^ F ) is non-special. Thatimplies the irreducibility. Indeed, first we show that an open dense sub-set of A^(d^F) is contained in V(d^m^k)'^ more precisely, it consists ofcurves of degree d having m nodes in a fixed position and k cusps in afixed position with fixed tangents. Then from the non-speciality we derivethat dimA^(d^F) = const, F G U, and that conditions imposed by fixedsingular points on curves of degree d are independent. Afterwards we rep-resent U as an open dense subset of the space of some linear bundle, whosefibres are A^(d,F), F e U, and whose base is an open dense subset ofSym^P2) x Sym^P^P2)), where P(TP2) is the projectivization of thetangent bundle of the plane.

BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE

238 E. SHUSTIN

The text is divided into five parts : in section 1 there are somepreliminary notions and results; in section 2 we present examples ofreducible varieties or ones without property T; in section 3 we constructirreducible curves in Ai(7i,P), where n < d; in section 4 we prove theproperty T; and in section 5 — the irreducibility.

1. PreliminariesHere we shall recall some notions and well-known classical results [3],

[14], and also present some simple technical results needed below. Namely,we introduce a certain class of linear systems of plane curves and showhow to compute their dimensions by means of linear series on curves.

Let S = (Dô ̂ W ê tne g^d^ ril^g of polynomials in threehomogeneous variables over the base field. We think of the space of planecurves of degree t as the projectivization P(S(^)). A linear system of planecurves of degree t is a subspace of P(S(^)). Let

i=^BmcEtÔ

be a homogeneous ideal, defining a zero-dimensional subscheme Z C P2.This ideal determines a sequence of linear systems A(t) = P(I(t)), t > 1.Denote this class of linear systems by C. In other words, these are linearsystems defined by linear conditions associated to finite many base points.It is well-known [3] that

(1.1) dimA(t) =dimP(S(t)) - degZ + z(A(t)),

where i(A(t)) > 0 is called the speciality index of A(t). If i(A{t)) = 0 thenthe linear system A(t) is called non-special. For a given ideal J, A(t) isnon-special when t is big enough (see [3]).

PROPOSITION 1.2. — Let A^.A^t) belong to C, and, for all t > 1,

A(t) c A\t).

If, for some n > 1, the system A(n) is non-special then A^n) is non-special.

Proof. — The systems A(^),A'(t) are non-special for t big enough.Take a straight line L G P(S(1)) not intersecting the zero-dimensionalschemes Z, Z ' associated to our linear systems. Let us embed the spaceP(S(n)) into P(S(t)), multiplying by L^-71. Then :

A(n) = A(t) n P(S(n)), A\n) = A\t) n P(S(n)).

TOME 122 — 1994 — ?2


So, the non-speciality of A(n) means the transversality of the intersectionof A(f) and P(S(n)) in P(S(f)). But this implies that A7^) and P(S(n))intersect transversally in P(S(t)), hence

coding (n),P(E(n))) = codin^A^), P(S(^))) =degZ/,

what is equivalent to the desired non-speciality. Q

From now on, divisor will always mean an effective Cartier divisor ona curve.

Let F be a reduced plane curve. For any divisor D on F and anycomponent H C F the symbol D\n means the restriction of D on H. Forany curve G the symbol Gp means the formal expression ̂ n(P)-P, wherethe sum is taken over all local branches of F and n(P) is the intersectionnumber of P and G. If P, G have no common components, Gp is thedivisor on F cut out by G, otherwise we admit infinite coefficients in theabove expression.

By D(F) we denote the double point divisor of the curve F. We omitits exact definition (see, for example, [14]), but only list the propertiesused in the sequel.

PROPOSITION 1.3 (see [14]).(1) The divisor D(F) can be expressed as

D(F)=^n(P)-P,

where P runs through all the local branches of F centered at singularpoints^ and the coefficients n{P) are positive integers. In particular^n(P) = 1 for both branches centered at a node, and n(P) = 2 for a branchcentered at a cusp.

(2) Let z be a singular point of the curve F and a non-singular pointof some curve G, then

(i) for any singular local branch P of F centered at z,

(G-P)(z)<n(P)+l,

(ii) for any pair Pi, ?2 of local branches of F centered at z,

(G'Pi)<n(Pi), or (G.P2)<n(P2).(3) If F is an irreducible curve of degree d and geometric genus g(F)

then :degP(P) = d(d - 3) + 2 - 2g(F).


240 E. SHUSTIN

(4) If a reduced curve G has no common components with F then :

D{FG)\F=D{F)^GF.

For any divisor D on F^ the symbol /^(n, D) denotes the linear systemof plane curves of degree n

{^ | <^F > D ^ D ( F ) } .

It is clear from the definition and PROPOSITION 1.3 that Ai(n, F), A^(n, F)belong to this class. Also these systems belong to C.

THEOREM 1.4 (Brill-Noether (see [14])).—If F is irreducible then curvesfrom /^(n, D) cut out on F the linear series \nLp — D — D(F)\^ where Lis a general straight line.

THEOREM 1.5 (Noether (see [14])). — Let Fi,..., F^ be different irre-ducible curves of degrees n\^..., 7^5 a^d F = F\ • • • F/c, deg F = d. Then :

k

(1.6) /:^(n,D)=^/:^(n+^-d,P|^)-Fr.-F,_iF,+i...Ffc.i=l

THEOREM 1.7 (Riemann-Roch for curves (see [3], [14])). — For anydivisor D on an irreducible curve F the dimension of the linear series \D\is

dim D\ = deg D - g ( F ) + i(D),

where i(D) is non-negative. If deg D > '2g(F) — 2 then i(D) = 0.

PROPOSITION 1.8. — For any reduced curve F of degree d < n,

(1.9) dim CF{n, D) ̂ ^ n{n + 3) - ^ deg D(F) - deg D.

The non-speciality of jCp^n^D) is equivalent to the equality in (1.9).

Proof. — Assume that F is irreducible. Representing Cp^n.D) as thespan of \nLp - D - D{F)\ and F ' P(S(n - d)), we obtain

dimCp^D) = dim[nL^ - D - D(F)\ +dimS(n-d),

hence according to THEOREM 1.7 and PROPOSITION 1.3,

dim£F(n,D) ^ nd - degD - degD(F) - g{F)

+ j(n-d+l)(n-d+2)

=nd- ^d(d-3)-l-degD- \ degD{F)

+ j(n-d+l)(n-d+2),

TOME 122 — 1994 — N° 2


which is equivalent to (1.9). Also we obtain that the equality in (1.9)means i{D) = 0. Therefore, for all t > d,

(1.10) codim(/:F(^),P(S(^))) = j degD(F) + degD.

On the other hand, for t big enough, Cp^t.D) is non-special. Comparingthis with (1.1) and (1.10), we get that the equality in (1.9) means thenon-speciality of Cp^ri, D).

If F is reducible, combine the previous computation with (1.6).

PROPOSITION 1.11. — Let F e V(d,m,k).(1) // G G Ai(n,.F) is reduced, then there is a divisor D on G of

degree < m + 2k such that, for all t ^ 1,

(1.12) A,(t,F)^CG(t,D).

(2) Let G G Ai(n,F) be irreducible, let S be a subset ofSmg(F), andlet H be a reduced curve containing S but not G. Let A^(t, F, S) be a linearsystem of curves ^ G Ai(^,F) such that S C Sing(^>), and ^ meets F ateach cusp from S with multiplicity ^ 5. Then there is a divisor D on GHsuch that

degD\G ^ m-\-2k, degD\^ ^ card(6' H K),

for each component K C H, and, for all t > 1,

A^t,F,S)^CGH(t,D).

Proof. — We will construct the divisor D = ̂ n(P) • P explicitly.(1) We have to find a divisor D on G such that any curve from

CG^.D) goes through each node and each cusp of F, and intersects atangent line to F at any cusp with multiplicity > 2.

Let z be a node of F. Since G e Ai (n, F), then G goes through z. If G isnon-singular at z we can put n(P) = 1 for the local branch P of G centeredat z. If G is singular at z then we can put n(P) = 0 for all local branchesof G centered at z, because in this case, according to PROPOSITION 1.3,curves from Ccit^ff) go through z.

Let z be a cusp of F. Analogously, G goes through z. If G is non-singularat z, then the local branch P of G at z is tangent to the tangent line Lto the curve F at z. Put n(P) = 2. Now, since any curve from /^(p, D)intersects P with multiplicity > 2, the same holds for L. If G is singular


242 E- SHUSTIN

at z, then either there is a singular local branch P of G centered at ^or there are at least two local branches Pi, ?2 of G centered at z. In^hefirst case we put n(P) = 2, in the second case we put n(Pi) = n(?2) = 1.According to PROPOSITION 1.3 any curve from CG^^D) is singular at z,and thereby intersects L with multiplicity > 2.

(2) We can obtain the second statement easily by combining theprevious arguments with the Noether theorem. D

2. Non-transversality and reducibilityThe upper bounds in the sufficient conditions (0.3), (0.4) are the best

possible as far as the exponent of d is concerned. The slightly modifiedclassical examples [15] presented below give an upper bound for theallowable coefficient of d2 in (0.3), (0.4).

THEOREM 2.1. — The set y(6j9,0,6p2) is reducible ifp = 1,2, and hascomponents with different dimensions ifp > 3.

Proof. — The case p = 1 is well-known [15]. Let p > 2. It is easy to seethat the curves

H=F^+Gl,

belong to V(6p, 0,6p2), where F^p, Gsp are general curves of degrees 2p, 3prespectively. A simple computation gives us :

dim{H € Y(6p, 0,6p2) H = F^ + GJp}w = i6p(6p + 3) - 12p2 + | (p - l)(p - 2).

According to [13], for p > 2, there is a component of V(6p,0,6p2) withdimension :

l6p(6p+3)-12p2.

If p ^ 3 we obtain at least two components of V(6p, 0,6p2) with differentdimensions.

Let p = 2. According to (0.5), V(12,0,24) is a T-variety, and hence hasdimension 42. According to (2.2^ curves H = F^ + Gj form a componentV of V(12,0,24). Assume that V == V(12,0,24).

Let J be an irreducible curve of degree 12 with 28 cusps constructedin [13]. Since V(12,0,28) is a T-variety (see (0.5)), we can smooth outany four cusps of J, preserving the others, by means of a variation of Jin the space P(S(12)). Indeed, since all 28 equisingular strata intersecttransversally at J, we can leave four of them by moving J along theintersection of the others. So we obtain that J belongs to the closure of V,

TOMB 122 — 1994 — N° 2


and hence to any set 534 of 24 cusps of J there correspond a quartic F^ anda sextic GQ^ passing through ^24. Distinct 24-tuples of cusps correspondto distinct quartics, because, according to Bezout's theorem, a quarticcannot contain more than 24 cusps of J. On the other hand, two 24-tuples 524, .§24 with 23 common cusps give quartics ^4, F^ with 23 commonpoints. Therefore ^4, F^ have a common component Ci of degree z =1, 2,or 3. If i = 3 then F^ = C^Ci, F^ = C^C[. Since €3 passes through atmost 18 cusps of J, then the straight lines Ci, C[ have at least 5 commonpoints, that means they coincide. The cases i =1 or 2 lead analogously tocontradictions, which prove that Y(12,0,24) is reducible.

THEOREM 2.3.— The set V(7p—3, 0,6p2) contains a component withoutproperty T when p > 3.

Proof. — Obviously, the curve H = Ap-^F^ + Bp-^G^ belongs toV(7p — 3,0,6p2), if Ap_3, F^p, Bp_3, G^p are general curves of degreesp — 3, 2p, p — 3, 3p respectively. The property T is equivalent to the non-speciality of Ai(7p—3, H). From THEOREM 1.5 it is not difficult to deducethat

Ai(7p - 3, H) = {^> ] ^> = R^F2? + S^G^p}

with arbitrary curves I?3p_3, 64^-3 of degrees 3p — 3,4j? — 3. Further, atrivial computation gives

dim Ai (7p-3,H) == j (7p - 3) • 7p - 12p2 + 1,

that means Ai(7j9 — 3,7^) is special.

COROLLARY 2.4. — The allowable coefficient at d2 in the right handside of (0.3) cannot exceed ^j, and in the right hand side of (0.4) cannotexceed j.

3. Main lemma

LEMMA 3.1. —For any curve F £ V{d,m,k) and real a ̂ (m-\-2k)/d2,there is an irreducible curve <I> € Ai(n, F), w/iere n = [(\/2a + 2/3)c?].

Proof. — Let 2 ^ 1 , . . . , Zm be the nodes of F^ and let w i , . . . , w^ be thecusps of F. Let /z be the minimal integer such that Ai(/i,F) -^ 0. Then,Ai(/^- 1,F) = 0 implies

m+2k> l(/^-l)(/l+2),

and hence

(3.2) /i< ^/2(m+2A;) < V2ad.


244 E. SHUSTIN

Take a general curve H c Ai(/i, F). Assume that H = H { 1 ' ' ' H^, whereH ^ , . . . , Hr are irreducible components of degrees /î,..., hr respectively.Since h is minimal,

max{%i , . . . ,%y} < 2.

We shall construct the curve ^ as follows. First we will construct, for eachs = 1,. . . , r, a curve Cs of degree £g < ishs + j d such that Cs does notcontain Hs and the curve

(3.3) R, ̂ H{1. • • ̂ -i1 C7,J%1 . • . H^

belongs to Ai(/i + ̂ - ishs, F). After that we obtain the desired curve în the form

G o H + G i R ^ + ' - ' + G r R r ,

where Go, G i , . . . , Gr are generic curves of suitable degrees.

The rest of the proof is divided into five steps : in steps 1, 2, 3 and 4 weconstruct the curves (7i, . . . , Cy, in the fifth step we construct the curve <I>.

Let us do the construction of C\. Let H^ pass through î , . . . ,2 ;p ,w i , . . . , W g and meet F at Wg+i , . . . ,Wg+i with multiplicities > 3. Letdegî = /ii. The Bezout theorem gives :

(3.4) 2p + 2q + 3t < h^d.

Step 1. — Assume %i = 1. Let us find a curve C\ passing through^ i , . . . , Zp, w i , . . . , Wq, meeting F at w^+i , . . . . Wq^ with multiplicities > 3,and not containing H^. This can be done under the following sufficientcondition on fi\ = deg C\

\ ̂ (î + 3) - \ (£, - h,) (î - /,i + 3) > p + q + ̂

which is equivalent to

P ^ I/, 3 , P + g + 2 t^1 > 9^1 — o + ———,————'" " h\

and, using (3.4), we can take

(3.5) ,̂.̂ ^^^< ,̂

5'̂ ej) .̂ — From now on assume zi = 2. First we look for a curve C[of degree £[ passing through z^..., Zp, meeting F at w^+i , . . . . Wg-^ with

TOME 122 — 1994 — ?2


multiplicities > 3 and not containing Hi. As in the first step we have thefollowing sufficient condition for the existence of such a curve

';>^-i^'and then we can take

(3.6) ^<^+^.

Step 3. — Assume that there is a curve H[ 7^ H\ of degree h[ < fai,passing through w i , . . . , Wg, and that h^ is the minimal such degree. Thisminimality implies :

(3.7) h[2 < 2q.

Then we put Ci = C[(H[}2. According to (3.6)

£i = degCi < 1^ + p+2t + 2h[.L hi

Hence, using (3.4), (3.7) and the initial assumption h[ < hi, it is easy tocompute

(3.8)

p < IT , , P+q+^t qt\ < ^ h\ + ———————d — — + 2/ii2 2p + 2q + 3t hi 1

< i/ii+ jd-—+2h[ < J^+2/î .1 ^ , 2 ^ _^ i o^' /- 2 .hi

êj? ^. — Now assume that Hi is the unique curve of degree < hi,passing through w i , . . . , Wq. In particular, that means

(3.9) î(/M+3) <q.

Let C[ be the curve of degree £[ constructed in step 2. Consider twosituations.

• Assume first q < h\. Then 2q < 2hi{2hi + 3)/2. That means theset M of curves of degree 2hi, meeting F at wi , . . . , Wq with multiplici-ties > 3, is infinite. The only curve in M containing Hi is H^. Now takeH[ C M different from J^2, and put Ci = C[H[. Here :

1. , P^î

£i=degCi < lhl+p——-^2hl.z h\


246 E. SHUSTIN

Finally, from (3.4), (3.9) we get :

(3.10) î < jd+2/i i .

• Now assume :

(3.11) 9 > ^ + 1 .Introduce the integer

h[ = max{;/ C N | h^(v - /ii) + 1 < q}.

According to (3.11), h[ ^ 2/ii. Denote by M the set of curves ofdegree h[ meeting F with multiplicity > 3 at each point W i , . . . , u^-, whereTT = /ii(/i^ - /ii) + 1. If G G M contains Jfi then G = Giî, and Gi goesthrough w i , . . . , w-^-, because

(Gi • F)(w,) = (G . F)(w,) - (H, . F)(w,) > 3 - 2 = 1, i = 1 , . . . , TT.

Hence Gi contains H\^ because these curves meet at TT > degH^ - degGipoints. Then there is a curve H[ G M not containing H\ as component,because the sufficient condition for this existence is

\ h[(h[ + 3) - 2(h^h[ - h^) + 1) > \ (h[ - 2h^)(h[ - 2/n + 3),which is equivalent to

3/ii > 2.Finally, let H'{ be a curve of the smallest degree h'[ meeting F at w^,% = hi{h[ — /î) + 2 , . . . , 9, with multiplicities > 3. By definition of h^ thenumber of these points is < h\ — 1. If h\ < 3 then, obviously, h'{ < h-^—l.If h^ > 4, since

^{hf{-l)(hf{^2)<2h^-2,

we have :

(3.12) h'[< |/ii.

The last inequality is true in the case h\ < 3 too. In particular, H'{ doesnot contain H^. Now put Gi = C[H[H'{. Here we have from (3.4), (3.6),and (3.12)

î - degGi < j/ii + p-^ + ̂ + ̂

(3.13) '̂ ^ '̂•i'-^

< j /» i+ jd+fa / l+/l / l /- j ^ -Ai )

< j / i i+ jd- |/i'i<2/ii+ jd,

because /i^ ^ 2/ii as it was mentioned above.

TOME 122 — 1994 — ?2


Step 5. — Inequalities (3.2), (3.5), (3.8), (3.10), (3.13) and the defi-nitions of n and a imply that the degree of any curve Rj, j == 1, . . . , r,defined by (3.3), is less than n. Now consider the linear system

(3.14) X^GoH + AiGi^i + • • . 4- XrGrRr , ( A o , . . . . A,) e P^

where Go,..., Gr are generic curves of positive degrees n - h, n - deg R^,. . . , n-deg Rr respectively. According to the construction of H, C - y , . . . , Cr\this is a subsystem of Ai(n, F). Also note that the curves Go,.. \, Gr donot go through base points of our linear system. Then we obtain'imme-diately from the Bertini theorem (see [3], [14]) that a generic member inthe linear system (3.14) is reduced and irreducible. Thereby we can takethis generic member as the desired curve <1>.

4. The property TLet F e V(d, m, k). As said above, the property T and the smoothness

of V(d, m, k) at F follow from :

PROPOSITION 4.1. — Under condition (0.3) the linear system Ai(d,F)is non-special.

Proof. — According to LEMMA 3.1, under condition (0.3) there isan irreducible curve ^ e Ai(n,F), n = [^2ao + jd]. According toPROPOSITION 1.11 there is a divisor D on <^> of degree

(4-2) degD<m+2k

such thatAi(p,F) 3£^{p,D), p> 1.

Therefore, according to PROPOSITION 1.2, it is enough to establish thenon-speciality of C^{d, D), which will follow from

(4-3) deg(^ - D(^>) -D) > 2g(^) - 2

where G <E £<p(d,D), and g(^) is the geometric genus of <^. Indeed, wehave by PROPOSITION 1.3 and (4.2) :

deg(G^ - DW - D) = nd - n(n - 3) - 2 + 2g(^>) - deg D

(4-4) > n(d - n + 3) - (m + 2k) + 2^) - 2

> n(d - n + 3) - aod2 + 2g(^) - 2.

Since n = [(y%o4- j)d] and OQ is the positive root of the equation

(^/2a+j)(^-^/2a)=a,


248 E. SHUSTIN

hencen(d-n+3) > aod2.

Then (4.4) implies (4.3) and completes the proof.

5. IrreducibilityWe prove the irreducibility along the plan mentioned in introduction.

PROPOSITION 5.1. — For any F G V(d,m,k), the intersection ofV(d,m,k) with A^(d,F) contains an open dense subset of A^(d,F), andconsists exactly of curves from V{d^ m, k) with the same nodes, and thesame cusps with the same tangents as F.

Proof. — Since F e A^d.F) and any curve G e A^d.F) is sin-gular at ^ i , . . . , ^ y n , then the Bertini theorem implies that almost allcurves in A-^{d^F) have nodes at ^ i , . . . , Z y ^ , and are non-singular out-side Sing(F). Consider the cusp wi € F. In some affine neighbourhoodof wi we fix an affine coordinate system (re, y) such that w\ = (0; 0), andy = 0 is a tangent to F at wi. Then in this neighbourhood, F is definedby polynomial

(5.2) Ay^^Bx3^- ^ A^V, AB ̂ 0,2i+3j>6

and can be locally parametrized analytically (see [3], [14]) by

(5.3) x=r2, y=\r3+0(r4).

To determine (G • F)(wi) we plug (5.3) into the affine equation

^a^y =0

of G, and then compute the order of vanishing at r = 0 (see [14]). Thuswe obtain for curves G C Aa (c?, F) that :

^oo = ̂ oi = ̂ io = ̂ n = ̂ 20 == 0 •

Since F c As^.F), almost all curves in A^d.F) have affine equationslike (5.2), that means they have a cusp at wi with the tangent y == 0. Nowto complete the proof we should note that any curve G G V(d, m, fc) withnodes ^ i , . . . , Zmi cusps w i , . . . , Wk and tangents T^F,..., T^F, belongstoA2(^F).

TOME 122 — 1994 — N° 2


PROPOSITION 5.4. — Under condition (0.4), V(d^m^k) contains anopen dense subset V consisting of curves F with non-special linear sys-tem A^d.F).

REMARK 5.5. — Even under condition (0.4), V(d,m,k) may containcurves G with special system Aa^, G). This holds for y(2p,3j?,0), p > 3(see [12]).

Proof of Proposition 5.4- — It is enough to prove the statement forany irreducible component Vo of V(d^m^k). For any curve G € Vo,let 1~i(G) denote the linear system of curves of the smallest degree /i,passing through Sing(G). Evidently, any H e 7Y(G) is reduced, and

(5.6) h =- degH < ̂ 2(m + k) < V2a^d.

Denote by V\ the set of curves G G VQ with maximal h. This is an opendense irreducible subset of Vo (see [3]). Now denote by V^ the set of curvesG G V\ with minimal dim'K(G). Similarly this is an open dense irreduciblesubset of V\. Then

w = \J H(G)GeV2

is irreducible as the image in P(S (h)) of the space of a projective bundlewith base V^ and fibres 7^(G), G G V^- Denote by Wo the set of curvesH G W with minimal number of irreducible components. It is irreducible,open and dense in W. In particular, that means all the curves H e WQdetermine the same sequence of degrees of their components (up topermutation). Moreover, if H runs through Wo, then any of its irreduciblecomponents K runs through some irreducible set

W(JQcP(S(degX)).

For H <E WoHT-^G) define N{H) to be ]̂ N(K, H), where K runs throughall components of H, and N(K, H) = card(J^ H Sing(G)). Denote by W^the set of curves H e Wo with minimal N{H). First, it is an open denseirreducible subset of Wo, and, second, for any component K of H 6 TVi,

N { K , H ) = mm N ^ K ^ H ' ) .K'^W{K)

At last, introduce

Vs = {G C V2 | H(G) H W^ ̂ 0}.

According to the above construction, this is an open dense subset of Vo.Now we will show that V^ contains an open subset consisting of curvessatisfying the conditions of PROPOSITION 5.4, in three steps.


250 E. SHUSTIN

Step 1. — Fix G e Vs and H € 7<(G) n W^. Show that any componentK of H oi degree 6 contains at most ^S(6 4- 3) points from Sing(G).

Indeed, let K contain ^6(6 + 3) + 1 points from Sing(G). Denote theset of these points by S^ and consider the linear system A^(d^G^S). Letus take an irreducible curve ^ C Ai(n,G), n = [(| + ^/2a\ )d], fromLEMMA 3.1. According to PROPOSITION 1.11, for all t > 1,

A3(t,G,^):)/:<^(t,D),where

(5.7) degD|<^m+2A;, degL^^ ^(6+3)+1.

We shall show that £^j<(d,D) is non-special. According to PROPO-SITION 1.8 and arguments from its proof, this is equivalent to

i((d - 6)L^ - DW - D^) = i((d - n)LK - D(K) - D^) = 0.

According to THEOREM 1.7 these equalities follow from :

(5.8) (d - 6)n - degD(^) - degD^ > 2g(^) - 2,

(5.9) 6{d - n) - deg D(K) - deg D\ K > 2g{K) - 2.

According to PROPOSITION 1.3 and (5.7), inequality (5.8) follows from :

m + 2k < n{d - 6 - n + 3).

This inequality can be easily deduced from the definition of n and (5.6),because ai = — satisfies the inequality :

ai < ( j +^/2o7)(|t -2^/2o7).

Analogously, by PROPOSITION 1.3 and (5.7) the inequality (5.9) followsfrom :

d-n^ j^.

This can be easily deduced from the definition of n and (5.6), because a\is the root of the equation

1 - ( J +V2a) = JV2a.

So, according to PROPOSITION 1.2, As^d.G^S) is non-special, andaccording to PROPOSITIONS 1.3 and 1.8

(5.10) dimA3(d,G,5') = ^d(d+ 3) - m - 2k - 2 • card 5.

TOME 122 — 1994 — ?2


If G runs through Vg, then S runs through some subset of

Sym^^+^P2),

thereby defining a morphism

^^-^Syni^3)/2^?2),

where 1/3 is the finite covering of V^ corresponding to different choicesof ^ 6(6 -h 3) + 1 points from the set Sing(G) D K (which might containmore than j 6(6 + 3) + 1 points). The tangent space to the fibre ̂ ^{S) atthe point (G, S) € ¥3 is contained in A^d.G.S) (see [2], [12]). Therefore(5.10) implies :

dim^-^) = dimYs - 2(|<^ + 3) + l),

hence v(V^) is dense in Sym6^3^2^1^2). Therefore there is a curveG <E ¥3 such that, for any set S of ^6(6 + 3) + 1 points in Sing(G) ft K,no more than ^6(6 + 3) points of S lie on a curve of degree 6. But thiscontradicts the definition of the set Vs and the initial assumption thatK D Sing(G) contains more than ^6(6 + 3) points, and thus completesthe proof.

Step 2. — Consider the linear system A^(d,G,Smg(G)). As in theprevious step, for any curve H £ "H (G) H TVi and all t > 1 we have

A3(t,G,Sing(G)) D C^H^D),

where D satisfies (5.7) for any component K C H^ hence A3(d, G, Sing(G'))is non-special.

Step 3. — As in the first step, the non-speciality of A^(d, (5, Sing((7)),G € ^3, implies that the image of V^ by the morphism

/2 : 13 ̂ Syn^+^P2), ^(G) = Sing(G),

contains an open dense subset U of Sym7714'^?2). According to [4], undercondition (0.4), for any Z from some open subset U ' C U the linearsystem A(d, Z) of curves of degree d, having multiplicity > 3 at eachpoint z e Z, is non-special. It is easy to see that for any G G V(d,m,k)and t > 1 :

A2(^G)DA(^Sing(G)) .

Therefore A^(d^ F) is non-special for any curve F e /^(L^) D ¥3. []

BULLETIN DE LA SOCIETE MATHEMATIQUE DE PRANCE

252 E. SHUSTIN

Now we can finish the proof of the irreducibility of V{d, m, k), showingthat V is irreducible. To any curve F € V we assign the set Sing(F) andthe set of tangents at its cusps. Thereby we obtain a morphism

^ : V —> Sym^P2) x Sym^TP2)),

where P(rP2) is the projectivization of the tangent bundle of the plane.According to PROPOSITION 5^1 any fibre of TT is an open subset of somelinear system A.^(d, F), F € V, hence is irreducible. The non-speciality ofthese linear systems, PROPOSITIONS 1.2 and 1.8 imply immediately thatall the fibres have the same dimension :

^d(d+ 3) - 3m - 5k = dimV - dim (Sym^P2) x Sym^P^P2))).

Finally, this equality means that 7r(y) is dense in

Sym^P2) x Sym^PCTP2)),

hence is irreducible. This completes the proof.

BIBLIOGRAPHY

[1] GREUEL (G.-M.) and KARRAS (U.). — Families of varieties withprescribed singularities, Compositio Math., t. 69 (1), 1989, p. 83-110.

[2] HARRIS (J.). — On the Severi problem, Invent. Math., t. 84, 1985,p. 445-461.

[3] HARTSHORNE (R.). — Algebraic geometry. — New York, Springer,^77-

[4] HIRSCHOWITZ (A.). — Une conjecture pour la cohomologie desdiviseurs sur les surfaces rationelles generiques, J. Reine Angew.Math., t. 397, 1989, p. 208-213.

[5] HIRZEBRUCH (F.). — Singularities of algebraic surface and character-istic numbers, Contemp. Math., t. 58, 1986, p. 141-155.

[6] IVINSKIS (K.). — Normale Flachen und die Miyaoka-KobayashiUngleichung. — Diplomarbeit, Bonn, 1985.

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[7] KANG (P.-L.). — On the variety of plane curves of degree d with6 nodes and k cusps, Trans. Amer. Math. Soc., t. 316 (1), 1989,p. 165-192.

[9] KANG (P.-L.). — A note on the variety of plane curves with nodesand cusps, Proc. Amer. Math. Soc., t. 106 (2), 1989, p. 309-312.

[9] SEVERI (F.). — Vorlesungen uber algebraische Geometric (AnhangF).— Leipzig, Teubner, 1921.

[10] SHUSTIN (E.). — Smoothness and irreducibility of varieties of planesingular curves, Arithmetic and geometry of varieties, Kuibyshev,Kuibyshev Univ. Press, 1989, p. 132-145 (Russian).

[11] SHUSTIN (E.). — Geometry of discriminant and topology of algebraiccurves, Proc. Intern. Congress Math., Kyoto, Aug. 21-29, 1990, vol. 1,Kyoto, 1991, p. 559-567.

[12] SHUSTIN (E.). — On manifolds of singular algebraic curves, SelectaMath. Soviet., t. 10 (1), 1991, p. 27-37.

[13] SHUSTIN (E.). — Real plane algebraic curves with prescribedsingularities, Topology, t. 32 (4), 1993, p. 845-856.

[14] VAN DER WAERDEN (B.L.). — Einfiihrung in die algebraischeGeometric. — 2nd ed., Berlin, Springer, 1973.

[15] ZARISKI (0.). — Algebraic surfaces. — 2nd ed., Heidelberg, Springer,Wi-


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