Counting singular plane curves via Hilbert schemes

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Advances in Mathematics 179 (2003) 38–58

Counting singular plane curves via Hilbertschemes

Heather Russell*

Department of Mathematics, Oklahoma State University Stillwater, Stillwater, OK 74078, USA

Received 6 July 2001; accepted 14 October 2002

Communicated by Michael J. Hopkins

Abstract

We give a method of counting the number of curves with a given type of singularity in a

suitably ample linear series on a smooth surface using punctual Hilbert schemes. The types of

singularities for which our results suffice include the topological type with local equation

xa þ yb withpap3b:We work out the example of curves with the analytic type of singularitywith local equation x2 þ yn for 1ono9:r 2002 Elsevier Science (USA). All rights reserved.

MSC: 14N10

Keywords: Hilbert scheme; Singularities

1. Introduction

Consider the following question. Given a suitable linear series L on a smoothsurface S; how many curves in L have a given analytic or topological type ofsingularity? By ‘‘suitable’’ linear series with respect to a type of singularity, we meanthat there are finitely many curves with the singularity in the linear series and theircodimension is maximal. Our approach to this question is to express the answer as aChern number of a vector bundle over a compactification of a space linearizing thecondition of having the singularity. For example, the condition of having a cuspalong a given tangent direction at a given point is linear in the sense that curves in a

ARTICLE IN PRESS

*Fax: +405-744-8275.

E-mail address: [email protected].

0001-8708/03/$ - see front matter r 2002 Elsevier Science (USA). All rights reserved.

PII: S 0 0 0 1 - 8 7 0 8 ( 0 2 ) 0 0 0 2 6 - 9

linear series spanned by two curves with the condition also have the condition. Thus,the projectivized tangent bundle PTðSÞ linearizes the condition of having a cusp inS: Note that the closure of the condition of having a cusp in along a given directionis the condition of containing a particular subscheme of S isomorphic to

SpecðR=ðx2; xy2; y3ÞÞ; where R is the ring K ½½x; y�� and K is the field of definitionof S: Generalizing this, the spaces we will use to linearize our conditions will be ofthe form

UðIÞ ¼ faAHilbdðSÞ : aDSpecðR=IÞg

for ideals I of some finite colength d in R: Section 3 is devoted to this correspondencebetween ideals and types of singularities and other conditions on curves. The space

UðIÞ has a natural compactification CðIÞ; its closure in HilbdðSÞ: Letting L be theline bundle corresponding to the divisor of a section of L; CðIÞ admits the vectorbundle VLðIÞ defined in Section 4 with the property that sections ofL give sectionsof the VLðIÞ vanishing exactly over those points corresponding to the data of thesingularity in question of the curve corresponding to the section. Thus the number ofcurves having the given types of singularity is the number of places a set of sectionsof VLðIÞ coming from a basis of sections ofL become dependent. This is the Chernnumber of L: In order to express this Chern number in terms of the divisor D of asection of L and the Chern classes of the tangent bundle of S; one would like toknow the Chern polynomial of VLðIÞ pulled back to some space with a known Chowring, so that the relations in the Chow ring can be used to find the degree of the

relevant Chern class. In the case that I can be constructed from the ideals ðx; y3Þ andðx; yÞ or from ðx; y2Þ and ðx2; yÞ by taking sums, products, and images under theFrobenius morphism if we are working over positive characteristic we can both findthe Chow ring of a space dominating CðIÞ and the Chern classes of VLðIÞ in terms ofthis Chow ring. However, if I is constructed from the ideals ðx; y4Þ and ðx; yÞ orðx; y3Þ and ðx2; yÞ; we can find the Chow ring of a space dominating CðIÞ; but do nothave an algorithm for finding the Chern classes of VLðIÞ: However, by slightly adhoc means one can sometimes or possibly always find these Chern classes. In the lastsection we give examples of both enumerative problems solved solely by previousresults and enumerative problems solved by a mixture of previous results and ad hocmeans.Similar questions are dealt with, for example, in [1] and [3]. What distinguishes our

approach is that by treating only singularities supported at a point, we can deal withmore complicated types of singularities through Hilbert schemes.

2. Preliminaries

The foundation of this paper has been built up in [4,5]. We recall some definitionsand notation.

ARTICLE IN PRESSH. Russell / Advances in Mathematics 179 (2003) 38–58 39

We will use the following short hand for denoting monomial ideals. Given asequence of positive integers s1;y; sr; we will let Iðn1;y; nrÞ denote the idealðxr; xr1yn1 ; xr2yn1þn2 ;yÞ:Moreover, given a sequence of monomial ideal, I1;y; Ir

we let

UðI1;y; IrÞ ¼ fða1;y; arÞAUðI1Þ ? UðIrÞ: (pAX

and

j :R *- #OX ;p with jðI1;y; IrÞ ¼ ða1;y; arÞg

and CðI1;y; IrÞ be its closure in the appropriate product of Hilbert schemes. We willsay that CðI1;y; IrÞ is an alignment correspondence with interior UðI1;y; IrÞ:Moreover, we will say that the measuring sequence of I1;y; Ir is A1; A2 where A1(respectively, A2) is the ideal generated by images of x (respectively, y) underautomorphisms of R fixing y (respectively, x) and sending the Ij’s to themselves. In

accordance with [4], although not [5], we let GðI1;y; IrÞ be the group ofautomorphisms of R sending the Ij’s to themselves.

3. Conditions on curves

Definition. Say that a curve C has the condition corresponding to an ideal I if C

contains a subscheme corresponding to a point of CðIÞ:

We will use the following two lemmas to identify conditions corresponding toideals.

Definition 3.1. Say that a curve C is generic with the condition corresponding to anideal I if I imposes independent conditions on the complete linear series containingC and C is generic among curves in this linear series with this condition.

Lemma 3.1. Given an ideal I of finite colength in R; the condition on the proper

transform of a generic curve with the condition corresponding to I with respect

to the blow up of S at the singular point of the curve is the condition corresponding to

the quadratic transform of I as defined in [7]. In particular, given a sequence of

positive integers n1;y; nr; the condition corresponding to Iðn1;y; nrÞ is the

closure of the condition of having an rth order point such that if y is a local coordinate

for the exceptional divisor of the proper transform of a generic curve C with the

condition, then the condition on the proper transform corresponds to the ideal Iðn1 1;y; nr 1Þ:

Proof. The lemma can be verified by direct computation in coordinate patches. &

ARTICLE IN PRESSH. Russell / Advances in Mathematics 179 (2003) 38–5840

Lemma 3.2. The condition corresponding to the integral closure of an ideal I is the

closure of the condition of having the topological type of singularity of a generic curve

with the condition corresponding to I :

Proof. By [7] (Proposition 5, p. 381) the quadratic transform of integral closure of I

is the integral closure of the quadratic transform of I : Thus the lemma follows fromLemma 3.1 and induction on the colength of I : &

Theorem 3.1. Let I be an ideal with measuring sequence at most ðx; y3Þ; ðx; yÞ(respectively ðx; y2Þ; ðx2; yÞ). Then the boundary of CðIÞ is equal to the space CðJÞwhere

J ¼ limt-N

gðtÞðIÞ

and gðtÞ is the automorphism of R sending x to x þ ty2 and fixing y (respectively, fixing

x and sending y to y þ tx).

Proof. The automorphisms of the form gðtÞ form a set of coset representatives ofGððx; y2Þ=GðI ; ðx; y2ÞÞ: Therefore, the boundary of the fiber of CðI ; ðx; y2ÞÞ overCððx; y2ÞÞ is the fiber of CðJ; ðx; y2ÞÞ over Cððx; y2ÞÞ with respect to the projectionmap. Hence projecting the space CðI ; ðx; y2ÞÞ to CðIÞ; the proposition follows. &

Definition 3.2. Given an ideal I as in Theorem 3.1, we will say that J is thedegeneration ideal of I : The degeneration ideal is then the unique ideal, up topermuting variables, that I degenerates to.

Definition 3.3. We will call the codimension of curves with the condition given by anideal I in a linear series on which I imposes independent conditions the codimension

of the condition, denoted codðIÞ:

Lemma 3.3. The codimension of the condition corresponding to I satisfies

codðIÞ þ dimðCðIÞÞ ¼ colðIÞ þ eðIÞ;

where eðIÞ is the dimension of the locus of subschemes in CðIÞ contained in a generic

curve with the condition given by I :

Proof. Let L be a linear series for which I imposes independent conditions.Consider the incidence correspondence

fðC; aÞAL CðIÞ: aCCg:

Equating sums of the dimension of the base and fiber with respect to the twoprojection maps, the lemma follows. &


Lemma 3.4. The codimension of the condition of having an analytic type of

singularity or a degeneration is the dimension of the versal deformation space of that

singularity.

Proof. The versal deformation space to a singularity can be naturally identified withthe normal space to the tangent space of the locus of curves in a linear series suchthat there is an ideal imposing independent condition on the linear series and genericmembers have the given type of analytic singularity. &

Example 3.1. Let n1;y; nr be a sequence of increasing positive integers. Then fromLemma 3.1 one can see that the condition given by the ideal Iðn1;y; nrÞ is the closureof the topological condition with Enriques diagram a succession of r free vertices ofdecreasing weights such that there are exactly ni vertices of weight at most r þ 1 i:

Example 3.2. A generic curve with the condition given by the ideal ðxa; ybÞ has thesingularity of topological type xa þ yb: The integral closure of the ideal ðxa; ybÞ is theideal Iða; bÞ generated by monomials xcyd with 3c þ dXb: Hence this ideal gives

the closure of the condition of having the singularity of topological type xa þ yb

and has measuring sequence ðx; y½ba�Þ; ðx; yÞ:

Example 3.3. The condition given by the ideal Ið2; 2; 1; 1Þ is strictly in between theclosures of the conditions of having topological type and analytic type x4 þ y6: Thetopological condition is that of having a double cusp. If one blows up at the doublecusp of a curve, the proper transform will be a tacnodal curve with both branchestangent to the exceptional divisor. If one then blows up at this point of intersectionsone gets a quadruple point, with two branches corresponding to the propertransform of the proper transform and two branches corresponding to exceptionaldivisors. The additional condition corresponding to Ið2; 2; 1; 1Þ is that the fourtangent directions of these four branches have cross-ratio 1: Such curves are ofanalytic type x4 þ ax2y3 þ y6 for some constant a: The analytic type varies with a2:Unlike the analytic condition above given by a cross-ratio, the analytic condition

corresponding to a particular choice of a2 cannot be realized by the configuration ofpoints of intersection of components of the total transform of the curve after somenumber of blow-ups. It would be interesting to find a geometric way of visualizingsuch analytic conditions.

Example 3.4. Theorem 3.1 can be used to glean some information aboutdegenerations of singularities. Given an ideal I as in Theorem 3.1, the conditiongiven by I is that of containing either SpecðR=IÞ or SpecðR=JÞ; where J is thedegeneration ideal of I :

For example, degeneration ideal of Ið3; 3; 2Þ is the ideal Ið2; 2; 1; 1Þ: Thiscorresponds to the fact that the singularity of topological type x3 þ y8 can


degenerate to the singularity of topological type x4 þ y6: However, only thosecurves with the additional analytic condition described in the example above canoccur.The sequence giving a condition is not in general unique. For example, the

condition corresponding to 1; n for any positive integer n is that of being singular.Note that for n > 2; a curve with a node will contain two schemes corresponding to

points in Cððx2; xy; y2ÞÞ: The following proposition gives a criterion for when twoideals correspond to the same condition.

Proposition 3.1. Let I1 and I2 be ideals in R with I1PI2: If

codðI1Þ ¼ codðI2Þ;

then both ideals give the same condition.

Proof. Let L be the linear corresponding to a sufficiently high tensor power of anample line-bundle on S: For iAf1; 2g; let Gi be the incidence correspondence inL CðIiÞ as in Lemma 3.3. Then each Gi is a vector bundle over CðIiÞ and henceirreducible. Therefore the image of the projection map p1: Gi-L is irreducible.Since p1ðG1Þ contains p2ðG2Þ and these two images are both of the same codimensioninL; they must be equal. Suppose by way of contradiction that the GðI1Þ orbit of I2is not equal to I1: Then there is an element aAI1 that is not in the GðI1Þ orbit of I2:One can find a curve CAL with local equation at a point pAS equal to the image of

a under an isomorphism from R to #OS;p up to an element in the image of a high

power of the maximal ideal in R: Then C is in p1ðG1Þ but not p2ðG2Þ: It follows thatGðI1Þ orbit of I2 is I1 and hence that I1 and I2 correspond to the sameconditions. &

Note that one can have ideals corresponding to the same condition, neither ofwhich is contained in the other. However, both will be contained in the ideal ofminimal colength corresponding to the condition.

4. Vector bundles on alignment correspondences

For each positive integer d let

UCHilbdðXÞ X

be the universal family over HilbdðX Þ:Let

p :U-HilbdðXÞ


and

m :U-X

be the projection maps.

Definition 4.1. Given a line bundle L on S; let VLðIÞ denote the restriction ofðpÞ

*mnðLÞ to CðIÞ or by abuse of notation its pullback to any space mapping

to CðIÞ:

The fiber in VLðIÞ over a point in CðIÞ is the vector space of germs of sections of L

modulo those in the ideal corresponding to the point. IfL is a linear series such thatL is the line bundle corresponding to the divisor of a section, then global sections ofL give global sections of VLðIÞ: These sections of VLðIÞ vanish exactly over thepoints of HilbdðX Þ parametrizing subschemes of the divisor of the correspondingsection ofL: Hence such a section vanishes if and only if the curve has the conditioncorresponding to I :By the following lemma, to find the Chern polynomial of VLðIÞ; it is enough to

find the Chern polynomial of VðIÞ:

Lemma 4.1. Let A be a space with a morphism to CðIÞ: Then for any line bundle L on

S; we have an equality of Chern polynomials

cðVLðIÞÞ ¼ cðVðIÞ#ðLÞÞ

over A: Here, abusing notation, we use L and VðIÞ to denote their pullbacks to A:

The proof is left as an exercise for the reader.

Proposition 4.1. Let CðI1;y; IrÞ be an alignment correspondence such that there is a

monomial ideal I with

I2CICI1

and

dimðCðI1;y; Ir; IÞÞ dimðCðI1;y; IrÞÞ ¼ dimðI1=I2Þ 1:

If I has dimension 1 (respectively codimension 1) as a subspace of I1=I2; then the space

CðI1;y; Ir; IÞ is the projectivization of the vector bundle VðI1=I2Þ (respectively,

VðI1=I2Þn) over CðI1;y; IrÞ:

Proof. If I has dimension 1 (respectively, codimension 1) as a subspace of I1=I2; then

CðI1;y; Ir; IÞ has a natural embedding in PVðI1=I2Þ (respectively, PVðI1=I2Þn).Since both spaces are irreducible and of the same dimension, this embedding is anisomorphism. &


The Chow rings of the spaces Cððx; y2Þ; ðx; y3ÞÞ and Cððx; y2Þ; ðx2; yÞÞ will beparticularly useful for enumerative applications due to the fact that they are

universal fiberwise AutðRÞ-equivariant compactifications of the spaces Uððx; y3ÞÞand Uððx; y2Þ; ðx2; yÞÞ; respectively.The following lemma can be found in [2]. Although the basic idea of the proof we

give is the same, we include a proof because in the course of the proof we set up aframework that we will use later.

Lemma 4.2. Let Ii denote the ideal ðx; yiÞ: Let c1 and c2 denote the first and

second Chern classes of the cotangent bundle of S; respectively. Let h2 denote the

hyperplane class of the projectivization of the cotangent bundle of S: Let h3 and

h03 be the hyperplane classes of the spaces CðI2; I3Þ and CðI2; ðx2; yÞÞ as the

projectivization of the bundles VðI2=I1I2Þ and VðI1=I21 Þ; respectively. Their Chow

rings are given by

AðCðI2; I3ÞÞ ¼ AðBÞ½h3�=ðh3 þ 2h2 þ 2c1Þðh3 h2Þ

and

AðCðI2; ðx2; yÞÞ ¼ AðBÞ½h03�=ððh0

3Þ2 þ h0

3c1 þ c2Þ:

Proposition 4.2. Let let VðI=JÞ be a bundle of rank one defined on Cððx; y3ÞÞ(respectively, Cððx; y2Þ; ðy; x2ÞÞ; where I and J are monomial ideals such that there

quotient is generated by xayb as a vector space. Let xcyd be the monomial generating

the degeneration ideal of I over the degeneration ideal of J:Then we have

c1ðVðI=JÞÞ ¼ ah2 þ bðc1 þ h2Þ þ ða cÞðh2 h3Þ

(respectively, ah2 þ bðc1 þ h2Þ þ ða cÞðc1 þ h03 þ h2ÞÞ:

Proof. Let Ij be the ideal ðx; yjÞ: Let I 0 and J 0 be the maximum monomial ideals with

respect to inclusion with measuring sequence at most I3; I1 (respectively, I2; ðx2; yÞ)such that I 0 is generated over J 0 by xayb and the degeneration ideal of I 0 is generated

by xcyd over the degeneration ideal of J 0:

Suppose aXc: Let L be the line bundle VðI3=I1I2Þ (respectively, VðI1=ðx2; yÞÞ:Then there is a map

VðI2=I22 Þc#VðI1=I2Þb#Lac-VðI 0=J 0Þ:

By Proposition 2.1 of [5], it is an isomorphism.


Similarly, if cXa; letting L be the line bundle VðI2=I3Þ (respectively, Vððx2; yÞ=I21 Þ)the map

VðI2=I21 Þa#VðI1=I2Þd#Lca-VðI 0=J 0Þ

is an isomorphism.Thus if the lemma holds for the four line bundles that we called L in the previous

two paragraphs, it always holds. The bundles VðI3=I1I2Þ and Vððx2; yÞ=I21 Þ are thetautological bundles over the spaces C3 and C2;2; respectively and hence have firstChern classes h3 and h0

3; respectively. Thus applying the Whitney productformula together with the knowledge of the middle elements from Lemma 4.2 to thesequences

0-VðI3=I1I2Þ-VðI2=I1I2Þ-VðI2=I3Þ-0

and

0-Vððx2; yÞ=I21 Þ-VðI1=I21 Þ-VðI1=ðx2; yÞÞ-0

we see that these four line bundles also satisfy the lemma. &

5. Some chow rings

In this section, we give the Chow rings of some spaces that can be expressed asfiber bundles over the projectivized cotangent bundle of S with toric varieties asfibers. Given a ray r; we let vðrÞ denote the smallest integral point which r passesthrough.

Lemma 5.1. Given an exact sequence of vector bundles

0-V2-V1-V3-0

the projectivization of V2 inside of the projectivization of V1 has class

cmðOV1ð1Þ#V3Þ;

where m is the rank of V3:

Proof. The map from V1 to V3 gives a section of HomðOV1ð1Þ;V3Þ which vanishesprecisely on the scheme-theoretic image of V2: &

The above lemma was communicated to me by Mike Roth.


Lemma 5.2. Let Y be a toric variety of dimension 2 with corresponding fan D:Let D1 be a torus invariant divisor corresponding to a ray r1 in D: Let D0 and D2be the torus invariant divisors intersecting D1 corresponding to rays r0 and r2just clockwise and just counter-clockwise of r; respectively. Let vi be the smallest

integral point that ri passes through. Then the intersection multiplicity of D0and D1 is

D0D1 ¼1

v04v1

and the self-intersection number of D1 is given by

D21 ¼v04v2

ðv04v1Þðv14v2Þ:

Proof. In the Chow ring of Y ; one has the relation

XriAD

ðv � viÞDi ¼ 0

for any vector v: Taking v to be orthogonal to v2 and intersecting with D1 we get therelations

ðv04v2ÞD0D1 þ ðv14v2ÞD21 ¼ 0:

Similarly, taking v instead to be orthogonal to v1; we get the relation

ðv04v2ÞD2D1 þ ðv04v1ÞD21 ¼ 0:

Since D2 does not effect the intersection of D0 and D1; we can assume that D1 and D2intersect in a smooth point while calculating D0D1: Since this also implies v14v2 ¼ 1;this gives us the first statement of the lemma. The second statement then followsfrom some additional algebra. &

Before stating the main result of this section, we recall some definitions from [5].Given a sequence of ideals I1;y; Ir with measuring sequence mð4; 1Þ or mð3; 2Þ;recall that UðI1;y; IrÞ is naturally a fiber over the projectivized tangent bundle of S

with fiber isomorphic to Gððx; y2Þ=Gððx; y4ÞÞ and Gððx; y2Þ=Gððx; y3Þ; ðx2; yÞÞ;respectively. The normalization of the closure of this fiber is a toric variety. Wesay that the standard fan of this toric variety is the fan with a ray through ð1; 0Þcorresponding to the divisor corresponding to automorphisms sending x to x þ ty2


for some tAK and fixing y and a ray through ð0;1Þ corresponding to

automorphisms sending x to x þ ty3 and fixing y if the measuring sequence ismð4; 1Þ and fixing x and sending y to y þ tx if the measuring sequence ismð3; 2Þ: Moreover, we say that a ray in a standard fan is a bounding ray if itcorresponds to a boundary divisor with only one Gððx; y2ÞÞ fixed point. It wasproved in [5] that in all characteristics but 2; the rays through ð0; 1Þ and ð1; 2Þ arebounding rays if they occur and that there are no other bounding rays.In characteristic 2; ð0; 1Þ is again a bounding ray if it occurs, but any otherbounding ray must lie in the interior of the convex cone bounded by the rays throughð1; 2Þ and ð0;1Þ:

Theorem 5.1. Let Y be an AutðRÞ equivariant compactification of Uððx; y4ÞÞ(respectively, Uððx; y3Þ; ðx; y2ÞÞ) over the space B ¼ Uððx; y2ÞÞ: Let D be the

standard fan (as defined in [5]) corresponding to the fiber of Y over B: Label the

rays in D clockwise starting from the ray through ð1; 0Þ so that ith ray is

labeled ri1: Let ðni;miÞ be the point of smallest positive distance from the

origin in ri having integral coordinates. Let r þ 2 be the number of rays in D:For 1pipr; let Di be the boundary divisor corresponding to ri: If either the

characteristic of K is not 2 or there is no bounding ray in the interior of

the cone bounded by the rays through ð1; 2Þ and ð0;1Þ then the Chow

ring of AðYÞ is generated over AðBÞ by the classes of the boundary divisors

which we will also denote Di by abuse of notation and the relations are gene-

rated by

D2k ¼ skDkDk1 þ Dkðakþ1h2 þ bkþ1ðc1 þ h2ÞÞ

for 1okpr;

D2k ¼ skDkDkþ1 Dkðak1h2 þ bk1ðc1 þ h2ÞÞ

for 1pkor; and

DiDj ¼ 0

for ji jjX2 where sk is the self-intersection number of the fiber of Dk and

ðak; bkÞ ¼ ðmk nk; 2mk 3nkÞ

(respectively, ðak; bkÞ ¼ ðmk þ nk; 2mk þ nkÞÞ:

Proof. The Chow ring of A is generated over the Chow ring of Uððx; y2ÞÞ by theboundary divisors because the restrictions of the boundary divisors to the fibersgenerate the Chow rings of the fibers. If ji jjX2 and fi; jgaf0; r þ 1g then the


relation DiDj ¼ 0 follows from the fact that Di does not intersect Dj: To verify

the remaining relations, first we will show that Di is isomorphic to theprojectivization of any bundle VðI=JÞ such that I and J have measuring sequence

at most ðx; y2Þ; ðx; yÞ and I is generated over J by xai and ybi : Then we showthat if ior then the intersection of Diþ1 with Di is given by the projectivizationof the vector bundle VðI1=JÞ where I1 is generated over J by xai :Moreover, we showthat if i > 1; then the intersection Di1 with Di is given by the projectivization

of the vector bundle VðI2=JÞ where I2 is generated over J by ybi : Then an applica-tion of Lemmas 4.2 and 5.1 give us some of the relations. The remainingrelations will be verified through those Y that are projectivizations of certain vectorbundles.

The function f :A2 0-P1 given by

f ðða; bÞÞ ¼ ðami ; bniÞ

extends to a regular function on the fiber of Di which we will also call f : The map

j: Di-PVðI=JÞ

such that for a point p in the fiber of Di over ðx; y2Þ; if f ðpÞ ¼ ðs; tÞ then

jðpÞ ¼ ðsxai þ tybiÞ þ J

is well defined because it is independent of the choice of x and y:Moreover, it can beshown to be an isomorphism. If ior; the intersection of Di with Diþ1 restricted to a

fiber is f 1ð1; 0Þ where f is restricted to the fiber of Di over ðx; y2Þ: Hence, if i > 1

PVðI1=JÞ ¼ Diþ1Di

and if ior

PVðI2=JÞ ¼ Di1Di:

Let x be the hyperplane class of PVðI=JÞ: By Proposition 4.2

AðDiÞ ¼ AðBÞ½x�=ðx aih2Þðxþ biðc1 þ h2ÞÞ:

Lemma 5.1 gives us the relations

ðx aih2ÞDi ¼ DiDi1

for i > 1 and

ðxþ biðc1 þ h2ÞÞDi ¼ DiDiþ1

for ior: Although, there are no global divisors D0 or Drþ1; if ri is not a boundingray, there are global divisors DiDi71 Since the intersection of the fiber of Di with the

coordinate axes is then Gððx; y2ÞÞ invariant. So, we extend these equations to any i


such that ri is not a bounding ray. Taking the difference of the two equations weobtain

DiðDiþ1 Di1Þ ¼ Diðaih2 þ biðc1 þ h2ÞÞ:

Hence for 1oipr and ri1 not a bounding ray

Di1D2i ¼ Di1Diðai1h2 þ bi1ðc1 þ h2ÞÞ

and for 1pior and riþ1 not a bounding ray

Diþ1D2i ¼ Diþ1Diðaiþ1h2 þ biþ1ðc1 þ h2ÞÞ:

Since similar relations hold in the Chow ring of the fiber of Y over B; the Chowring of Y has relations of the form

D2i ¼ siDiDi1 þ DiZ1

for i > 1 and

D2i ¼ siDiDiþ1 þ DiZ2

for ior where si is the self-intersection number of the fiber Di over B and Z is thepullback of a class from B: Multiplying these two equations by Diþ1 and Di1;respectively, we see that unless ri is a bounding ray, if i > 1;

Z1 ¼ aiþ1h2 þ biþ1ðc1 þ h2Þ

and if ior;

Z2 ¼ ai1h2 bi1ðc1 þ h2Þ:

It remains to verify the relations involving D2i for ri a bounding ray. Since the

relations depend on the neighborhood of Di; it is enough to verify them for somespace Y for each possible pair of rays corresponding to adjacent boundary divisors,such that one of the rays is a bounding ray. Any such pair occurs for a space Yði; nÞgiven by the ith row of Table 1 as follows. The integer n must be at least i 1; exceptthat Yððn; 3ÞÞ is independent of n: Let the ideal Jk be as given by the entries in the ithrow of the respective column. We define the space Yðn; iÞ to be the projectivization ofthe vector bundle VðJ1=J3Þ over the base B0 given in the first column. By Theorem4.1, this is also the space obtained by superimposing B0 with the space CðJ0Þ: Thus ifi ¼ 3; 4 (respectively, i ¼ 1; 2) then Yðn; iÞ is a compactification of Uððx; y4ÞÞ(respectively, Uððx; y3Þ; ðx2; yÞÞ: The second entry in the ith row gives the rays in thefan of the cone of the fiber of Y ðn; iÞ over B corresponding to the two boundarydivisors. The first ray corresponds to the pullback of the boundary of B0 and thesecond to the projectivization of the sub-bundle VðJ2=J3Þ over B0: By Lemma 5.1,these two divisors have classes h4 þ c1ðVðJ1=J2ÞÞ and h3 h2 (respectivelyh3 þ h2 þ c1) where h4 is the hyperplane class of Yðn; iÞ: Using this to eliminate h3


and h4 in the relation from the base C3 or C2;2 and the relation

ðh4 þ c1ðJ1=J2ÞÞðh4 þ c1ðJ2=J3ÞÞ ¼ 0

we recover the relations in the statement of the theorem. &

Proposition 5.1. Keeping the notation of the proof of Theorem 5.1, let J0; J1; J2;and J3 be the ideals associated to the space Yði; nÞ as given in Table 1 and r1 and

r2 the two rays given in the ith column of the table. Let D be the divisor of

Yði; nÞ corresponding to the ray through ð1; 0Þ if i ¼ 2 and the ray through ð0; 1Þotherwise. Then

c1ðVðJ0=J3ÞÞ ¼ D þ c1ðJ1=J2Þ

and

c1ðVðJ1=J0ÞÞ ¼ D þ c1ðJ2=J3Þ:

Proof. The bundle VðJ0=J3Þ is the tautological bundle over Yði; nÞ and thus has firstChern class h4: Recalling from the proof of Theorem 5.1 that

Dðr2Þ ¼ h4 þ c1ðVðJ1=J2ÞÞ

and the fact that

c1ðJ1=J0Þ þ c1ðJ0=J3Þ ¼ c1ðJ1=J2Þ þ c1ðJ2=J3Þ;

the proposition follows. &

ARTICLE IN PRESS

Table 1

Table of projective bundles

Base Rays J0 J1 J2 J3

C3 ð1; nÞ; ð0; 1Þ 2;y; 2|fflfflffl{zfflfflffl}n

; 3; 0 2;y; 2|fflfflffl{zfflfflffl}nþ1

; 1 2;y; 2|fflfflffl{zfflfflffl}nþ2

2;y; 2|fflfflffl{zfflfflffl}n

; 3; 1

C2;2 ðn; 1Þ; ð1; 0Þ 1;y; 1|fflfflffl{zfflfflffl}n1

; 0; 3 1;y; 1|fflfflffl{zfflfflffl}n1

; 0; 2 1;y; 1|fflfflffl{zfflfflffl}nþ1

1;y; 1|fflfflffl{zfflfflffl}n

; 2

C3 ð1; 2Þ; ð0; 1Þ 4 3 1; 2 1; 3

C3 ð1; nÞ; ð0; 1Þ 2;y; 2|fflfflffl{zfflfflffl}n3



2;y; 2|fflfflffl{zfflfflffl}n2

; 3

H. Russell / Advances in Mathematics 179 (2003) 38–58 51

6. Examples

In this section we find the number Ni of curves in a suitable linear series L on a

surface S with the singularity with local equation x2 þ yi for i from 2 to 8:We will letD denote the divisor of a section of L and L the associated line bundle on S:Moreover, we will let Im denote the ideal ðx; ymÞ: The ideal

Bi ¼ ðx2; xy½ i2�; yiÞ

corresponds to this type of singularity in the sense of Section 3. The linear series Lwill be of projective dimension n 1; giving n independent sections of the bundleVLðBiÞ over the space CðBiÞ: Up to scaling, the linear combinations of these sectionswith zeroes are in bijection with curves in L with the given singularity. Thus thenumber of these curves is the number of the vector bundle VLðBiÞ: This Chernnumber can be found by finding the Chern class of dimension 0 of VLðBiÞ and thenusing the Chow ring of the space CðBiÞ or some other space dominating to find thedegree of this class. Up to i ¼ 6; this information is given by Lemma 4.2 andProposition 4.2. For i ¼ 7 and i ¼ 8; the relevant Chow rings are given by Theorem5.1, but we will have to use slightly ad hoc means to find the Chern classes we areinterested in.To find the relevant Chern classes, we will find the Chern classes of the successive

quotients in the sequence

VðI24 Þ-VðI24 þ I71 Þ-VðI23 Þ-VðI23 þ I51 Þ-VðI2I3Þ

-VðI22 Þ-VðI22 þ I31 Þ-VðI1I2Þ-VðI21 Þ-VðI1Þ

and then apply the Whitney product formula together with Lemma 4.1.To find the number N2; the number of nodes in a pencil of curves, we need only

work over the surface S: Thus N2 is equal to the second Chern class of VðI21 Þ#L:Since this Chern class is already expressed in terms of the Chern classes of the surfaceand the divisor D; no further substitution is necessary. This example as well as theexample of computing N3 are worked out in detail in [6]. To find N3 and N4; thenumbers of cuspidal and tacnodal curves in a suitable linear series L; we will workover the projectivized cotangent bundle of S: The Chern classes c3ðVðB3Þ#LÞ andc3ðVðB4Þ#LÞ come expressed in terms of divisors pulled back from S and the hyper-

plane class h2 of the cotangent bundle. Using the relation h22 þ c1h2 þ c2 ¼ 0 to makethese classes linear in h2; the numbers, N3 and N4 are the coefficients of h2 of theseclasses. Similarly, to find N5 and N6; we work over the space YðI2; I3Þ: The fourthChern classes of the bundles VðB5Þ#L and VðB6Þ#L come expressed in terms ofpullbacks of divisors on S; the pullback of the hyperplane class h2 of theprojectivized cotangent bundle of S and the hyperplane class h3 of CðI1; I2Þ: Usingthe relations in the Chow ring of CðI1; I2Þ to make these Chern classes linear in h3and h2 separately, the numbers N5 and N6 are the coefficients of h2h3 in these twoclasses.


Finding the numbers N7 and N8 is a bit trickier because we must find the Chernpolynomials cðVðB7Þ#LÞ and cðVðB8Þ#LÞ by slightly ad hoc means. In particular,we need to find the Chern classes of the line bundles VðI23=I3I4Þ; VðI3I4=B7Þ; andVðB7=B8Þ: We will use Table 2 to see where the maps

j1 :VðI3=I4Þ2-VðI23=I3I4Þ;

j2 :VðI2=I3Þ3-VðI23=I3I4Þ;

j3 :VðI4=I1I3Þ#VðI3=I4Þ-VðI3I4=B7Þ

and

j4 :VðI1I3=I1I4Þ#VðI3=I4Þ-VðB7=B8Þ

and then apply Porteous’s formula. We will work over the compactification Y ofUðI4Þ with boundary divisors corresponding to rays through points ð0; 1Þ; ð1; 4Þ;ð1; 3Þ; ð2; 5Þ and ð1; 2Þ in the standard fan of the fiber, since this is the smallestbundle over which all of the vector bundles we will use are defined. Somevector bundles are also defined over smaller spaces. The Chern classes overthe smaller spaces can be related the Chern classes over Y via the followinglemma.

Lemma 6.1. Let Z and Z0 be two toric varieties of dimension 2 with fans D and D0

such that D0 is a subdivision of D: With respect to the map from Z0 to Z compatible

with these fans, the pullback of a divisor D corresponding to a ray r1 in D is of

the form

Xi

aiDi;

where i indexes of the rays ri in D0 and the Di’s are the corresponding divisors. The

integers ai can be found as follows. If ri lies strictly between r1 and an adjacent ray r2 in

D; then if

vðriÞ ¼ l1vðr1Þ þ l2vðr2Þ

then ai is equal to l1: If ri ¼ r then ai ¼ 1: In any other case, ai ¼ 0:

Proof. It is enough to check the lemma in the case that D0 differs D by a singlesubdivision corresponding to a ray r3; because the coefficient of the divisor

corresponding to this ray is the same as the coefficient of this ray for any D0


ARTIC

LEIN

PRES

S

Table 2

Boundary ideals

ð0; 1Þ ð1; 4Þ ð1; 3Þ ð2; 5Þ ð1; 2Þ

0; 3 ðx þ ay2Þ þ I1I2 I21 I21 I21 I210; 4 I1I2 I1I2 I1I2 I1I2 ða2y2 bxyÞ þ I31 þ I221; 3 ðxy þ ay3Þ þ I22 I31 þ I22 I31 þ I22 I31 þ I22 I31 þ I224; 3 yðx þ ay2Þ2 þ I32 I32 þ I51 ðbx3 þ 2a3xy3Þ þ I21 I22 þ I51 I31 I2 ða2y4 2bxy3Þ þ I21 I22 þ I513; 4 I2ðxy þ ay3Þ þ I32 I21 I2 I21 I2 ð3b2x3 þ a5y4Þ þ I31 I2 I411; 4 I22 I22 ð2bx2 þ a3y3Þ þ I21 I2 I31 I314; 4 I32 ðbx3 þ a4y5Þ þ I21 I22 I21 I22 þ I51 I21 I22 þ I51 ðx2; ðbx 2ayÞ2Þ2 þ I51

H.

Ru

ssell/

Ad

van

cesin

Ma

them

atics

17

9(

20

03

)3

8–

58

54

containing r3:Writing the pullback of D as D1 þ qD3; we find q by observing that theself-intersection of divisors is the same as the self-intersection of their pullbacks. Forany divisor D2 corresponding to a ray r2 we have DD2 ¼ ðD1 þ qD3ÞD2: If q ¼ 0;then D2 does not intersect D3: Otherwise, using Lemma 5.2 and writing r3 ¼q1r1 þ q2r2 and using Lemma 5.2 we can deduce that q ¼ q1: This verifies thelemma. &

Each column of Table 2 corresponds to a boundary divisor of Y and each rowcorresponds to an ideal as labeled on the top row and left-most column, respectively.

The sequence m; n corresponds to the ideal x2; xym; ymþn: Let Jðv; IÞ be the entrycorresponding to the ray through v and ideal I : This ideal is the projection of theimage of I under the generic automorphism to the smallest plane in the Pluckerembedding of the fiber CðB; IÞ over B containing the boundary divisor correspond-ing to v:The map j1 drops rank on the boundary divisors Di such that

Jðvi; I3Þ2CJðv; I3I4Þ:

Hence by Porteous’s formula over Y we have

c1ðVðI23=I3I4ÞÞ ¼ 2c1ðVðI3=I4ÞÞ þ n1Dð1; 2Þ

¼ 6ðc1 þ h2Þ þ 2Dð0; 1Þ þ 6Dð1; 4Þ þ 4Dð1; 3Þ

þ 6Dð2; 5Þ þ ð2þ n1ÞDð1; 2Þ;

where n1 is a positive integer. Here we have found c1ðVðI3=I4ÞÞ through Proposition5.1 together with Lemma 6.1. Similarly, one can find the Chern classes of VðI4=I1I3Þand VðI1I3=I1I4Þ over Y in this way. Following the same steps, Porteous’s formulaapplied to the remaining ji’s yields

c1ðI23=I3I4Þ ¼ 6ðc1 þ h2Þ þ n2Dð0; 1Þ þ ðn3 þ 3ÞDð1; 4Þ þ ðn4 þ 3ÞDð1; 3Þ

þ 6Dð2; 5Þ þ 3Dð1; 2Þ;

c1ðI3I4=B7Þ ¼ 3c1 þ 2h2 þ n5Dð2; 5Þ

and

c1ðB7=B8Þ ¼ 7ðc1 þ h2Þ þ 2Dð0; 1Þ þ 6Dð1; 4Þ þ ð4þ n6ÞDð1; 3Þ

þ ð7þ n7ÞDð2; 5Þ þ 3Dð1; 2Þ:


Comparing the two expressions for c1ðVðI23=I3I4Þ we see that n1 ¼ 1: SinceCðI2; I23 ;B7Þ does not have a boundary divisor Dð2; 5Þ; by Lemma 6.1 the coefficientof Dð2; 5Þ in the first Chern class of VðI23=B7Þ as a vector bundle over Y is the sum of

the coefficients of Dð1; 3Þ and Dð1; 2Þ: By the Whitney product formula,

c1ðVðI23=B7ÞÞ is the sum of c1ðVðI23=I3I4ÞÞ and c1ðVðI3I4=B7ÞÞ; we obtain n5 ¼ 1:Similarly, since CðI2; I23 ; I24 Þ does not have boundary divisors Dð1; 3Þ and Dð2; 5Þ; thecoefficient of Dð1; 3Þ in c1ðVðI23=I24 ÞÞ as a class in Y is half the sum of the coefficient

of Dð1; 4Þ and the coefficient of Dð1; 2Þ: Moreover, the coefficient of Dð2; 5Þ is halfthe sum of the coefficients of Dð1; 4Þ and three times the coefficient of Dð1; 2Þ: By theWhitney product formula, we have

c1ðVðI23=I24 ÞÞ ¼ c1ðVðI23=I3I4ÞÞ þ c1ðVðI3I4=B7ÞÞ þ c1ðVðB7=I24 ÞÞ:

Thus it follows that n6 ¼ n7 ¼ 1: The first Chern classes we have found or used alongthe way are listed in Table 3. The leftmost entry m1;m2;m3;m4 of each row signifiesthat the remaining entries in that row are the coefficients of the divisors listed at the

top of each column in the Chern class c1ðVððx2; xym1 ; ym1þm2ÞÞ=ðx2; xym

3 ; ym3þm4ÞÞover Y :Having found all relevant Chern classes, using the Whitney product formula

together with Lemma 4.1, we can find the Chern classes c5ðVðB7Þ#LÞ andc5ðVðB7Þ#LÞ: It remains to calculate the degrees of these Chern classes using therelations for the Chow ring of Y given in Theorem 5.1. Using these relations, theseclasses can be made free of squares of boundary divisors and linear in h2: Therespective degrees of the Chern classes are then the sums of the coefficients of termsof the form h2DiDiþ1: Our results are summarized below

N2 ¼ 6D2 þ 4Dc1 þ 2c2;

N3 ¼ 12D2 þ 12Dc1 þ 2c21 þ 2c2;

N4 ¼ 50D2 þ 64Dc1 þ 17c21 þ 5c2;

N5 ¼ 180D2 þ 280Dc1 þ 100c21;

N6 ¼ 630D2 þ 1140Dc1 þ 498c21 60c2;

N7 ¼ 2128D2 þ 4368Dc1 þ 2232c21 424c2

and

N8 ¼ 7272D2 þ 16544Dc1 þ 9548c21 2148c2:


In the special case when our surface is a hypersurface of P3 of degree d and thecurves in our linear series are cut out by hyperplanes, we have the following.

N2 ¼ 2d3 12d2 þ 34d;

N3 ¼ 4d3 36d2 þ 104d;

N4 ¼ 22d3 220d2 þ 608d;

N5 ¼ 100d3 1080d2 þ 2900d;

N6 ¼ 438d3 4884d2 þ 12798d;

N7 ¼ 1808d3 20528d2 þ 52768d;

N8 ¼ 7400d3 84336d2 þ 213328d:

Acknowledgments

I would like to thank Karen Chandler, Joe Harris, Tony Iarrobino, AnthonyKable, Steve Kleiman, Suresh Nayak, Dipendra Prasad, Mike Roth, Nilabh Sanat,

ARTICLE IN PRESS

Table 3

First Chern classes

c1 h2 Dð0; 1Þ Dð1; 4Þ Dð1; 3Þ Dð2; 5Þ Dð1; 2Þ

1,1/2,2 2 2

1,2/2,1 1

2,1/2,2 3 3

2,2/2,3 4 4 2 2 4 2

2,3/3,2 2 1

3,2/3,3 5 5 2 2 4 2

3,3/3,4 6 6 2 6 4 6 3

3,4/4,3 3 2 1

4,3/4,4 7 7 2 6 5 8 3

0,2/0,3 2 2 1 1 2 1

0,3/1,2 1 1 1 2 11,2/1,3 3 3 1 1 2 1

0,3/0,4 3 3 1 3 2 3 1

0,4/1,3 1 1 3 2 3 11,3/1,4 4 4 1 3 2 4 2

H. Russell / Advances in Mathematics 179 (2003) 38–58 57

Jason Starr, Ravi Vakil, Joachim Yameogo, and many others for their generoushelp. I would also like to thank the referee for several excellent suggestions.

References

[1] S. Colley, G. Kennedy, The enumeration of simultaneous higher-order contacts between plane curves,

Compositio Math. 93 (2) (1994) 171–209.

[2] A. Collino, Evidence for a conjecture of Ellingsrud and Str�mme on the Chow ring of HilbdðP2Þ;Illinois J. Math. 32 (1988) 171–210.

[3] S. Kleiman, R. Piene, Enumerating singular curves on surfaces, Algebraic Geometry: Hirzebruch 70

(Warsaw 1998), Contemparary Mathematics, Vol. 241, American Mathematical Society, Providence,

RI, 1999, pp. 182–192.

[4] H. Russell, Alignment Correspondences (preprint).

[5] H. Russell, Toric varieties in Hilbert schemes (preprint).

[6] R. Vakil, A beginner’s guide to jets (preprint).

[7] O. Zariski, P. Samuel, Commutative Algebra, Vol. II, Graduate Texts in Mathematics, Vol. 29,

Springer, New York, Heidelberg, 1975.


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