Structure of the cycle mapfor Hilbert schemes of families of nodal curves

Structure of the cycle map for Hilbert schemes of families of nodal curves

Abstract.

We study the relative Hilbert scheme of a family of nodal (or smooth) curves, over a base of arbitrary dimension, via its (birational) cycle map, going to the relative symmetric product. We show the cycle map is the blowing up of the discriminant locus, which consists of cycles with multiple points. We determine the structure of certain projective bundles called node scrolls which play an important role in the geometry of Hilbert schemes.

Key words and phrases:
Hilbert scheme, cycle map
14N99, 14H99

Introduction

In the classical (pre-1980) theory of (smooth) algebraic curves, a dominant role is played by divisors– equivalently, finite subschemes– and their parameter spaces, i.e. symmetric products. Notably, one of the first proofs of the existence of special divisors [6] was based on intersection theory on symmetric products, developed earlier by Macdonald [10]. In more recent developments however, where the focus has been on moduli spaces of stable curves, subscheme methods have been largely absent, replaced by tools related to stable maps and their moduli spaces (see [18] for a sampling stressing Vakil’s work). Our purpose in this paper (and others in this series) is to develop and apply global subscheme methods suitable for the study of stable curves and their families, aiming eventually, inter alia, to extend Macdonald’s theory to the case of families of curves with at most nodal singularities.

In this paper we always work over the complex numbers. Fix a family of curves given by a flat projective morphism

 π:X→B

over an irreducible base, with fibres

 Xb=π−1(b),b∈B

which are nonsingular for the generic and at worst nodal for every . For example, could be the universal family of automorphism-free curves over the appropriate open subset of , the moduli space of Deligne-Mumford stable curves. Consider the relative Hilbert scheme

 X[m]B=Hilbm(X/B),

which parametrizes length- subschemes of contained in fibres of . This comes endowed with a cycle map (also called ’Hilb-to-Chow’– in this case, ’Hilb-to-Sym’– map) to the relative symmetric product

 cm:X[m]B→X(m)B.

See §1 for a review. Because may be considered ’elementary’ (though it’s highly singular- see [14]), is a natural tool for studying . The structure of is the object of this paper. Our first main result is the following theorem which was announced with a sketch of proof in [13], where some applications are given as well.

Blowup Theorem.

is equivalent to the blowing up of the discriminant locus

 Dm⊂X(m)B,

which is the Weil divisor parametrizing nonreduced cycles.

In particular, we obtain an effective Cartier divisor

 2Γ(m)=c−1m(Dm)

so that can be identified with the natural polarization of the blowup. In fact, we shall see that also exists as a Cartier divisor, not necessarily effective, and the dual of the associated line bundle, i.e. , will be (abusively) called the discriminant polarization (though ’half discriminant’ is more accurate); we will also refer to itself sometimes the discriminant polarization. We emphasize that the Blowup Theorem is valid without dimension restrictions on . As suggested by the Theorem, the discriminant polarization encodes the additional information in Hilb vis-a-vis the unwieldy Sym and so, unsurprisingly, plays a central role in subsequent developments of geometry and intersection theory on the Hilbert schemes .

The proof of the Blowup Theorem occupies §§2-4 and may be outlined as follows.

(i) A preliminary reduction is made to the local case (§2);

(iI) we construct an explicit local model for the relative Hilbert scheme (§3);

(ii) we construct an ideal in the relative Cartesian product, whose syzygies correspond, essentially, to the defining equations of the pullback of over the Cartesian product; this yields a map from the blowup of to 4);

(iv) using the local analysis, it is shown that is an isomorphism and that is the ideal of the ordered discriminant (big diagonal);

(v) consequently descends to an isomorphism from the blowup of the ideal of the discriminant to .

The usefulness of the local model extends far beyond the Blowup Theorem; in particular, it yields information about the singularity stratification, of , which may be defined as follows. Let be a collection of distinct, hence disjoint, relative nodes of the family, each living in the total space over its own boundary component, and let be integers. Set

 Sm,n.(θ.;X/B)={z:cm(z)≥∑niθi}⊂X[m]B

This is mainly interesting when all . In this case, we construct a surjection

 ⋃1≤ji≤ni−1,∀1≤i≤rFm,n.j.(θ.;X/B)↠Sm,n.(θ.;X/B)

where each , called a node polyscroll (or node scroll, when ), is a -bundle over the smaller Hilbert scheme , where denotes the blowup (=partial normalization) of in , defined over the intersection of the boundary components corresponding to the . The fibre parameter of -th factor of the node polyscroll encodes a sort of higher-order (more precisely, -st order) ’slope’, locally at the -th node, and these together constitute the additional information contained in the Hilbert scheme beyond what’s in the symmetric product.

In the following section §6 we give an analogue of the blowup theorem in the case of flag-Hilbert schemes, which are often important in inductive arguments and procedures.

Our next main results (see Theorems 3, 5) determine the structure of node polyscrolls as -bundles. In fact, the disjointness of the nodes (in the total space) implies that the factors ’vary independently’, which allows us to reduce to the case of node scrolls, i.e. .

Actually, what’s essential for the enumerative theory of the Hilbert scheme, as studied e.g. in [16], and in which node scrolls play an essential role, is the structure of the node scroll as a polarized bundle, that is, the rank-2 vector bundles so that there is an isomorphism , under which the canonical polarization on associated to the projectivization corresponds to the restriction of the discriminant polarization on . To state the result (approximately), denote by the node preimages on , and by the relative cotangent spaces to along them, and by , for any divisor on , the ’norm’ of , considered as a divisor on .

Node Scroll Theorem.

There is a polarized isomorphism

 Fm,nj(θ)=P(O(−Dnj(θ))⊕O(−Dnj+1(θ)))

where

 Dnj(θ)=−(n−j+12)ψx−(j2)ψy+(n−j+1)[m−n]∗θx+j[m−n]∗θy+Γ[m−n].

This result, and its polyscroll analogue, reduce intersection theory on polyscrolls to that of the Mumford tautological classes, about which a great deal is now known thanks to the work of Witten, Kontsevich, Faber and many others (see e.g. [18] and references therein). The Node Scroll Theorem is one of the main ingredients of a complete ’Hilbert- tautological’ intersection calculus, developed in [16], which allows us to extend the intersection theory and enumerative geometry of a single smooth curve, as developed notably by Macdonald [10] and presented in [2], to the case of families of curves with at most nodal singularities, extending work of Cotteril [3] in low degrees. As described in [16], this intersection calculus has now been implemented on the computer, in the form of a Java program called macnodal [9], due to Gwoho Liu and available from the author’s web page. See also [12] for an application to the class of the closure of the hyperelliptic class in .

Acknowledgments  I thank Mirel Caibar for asking some stimulating questions early on, and the referee for many helpful, detailed comments and suggestions.
Convention  In this paper we always work over .

Part I Blowup theorem and discriminant polarization

1. Review of cycle map

1.1. Norms and multisections

Let be a finite, flat, degree- morphism of algebraic -schemes, corresponding to a sheaf of - algebras that is locally -free of rank . The action of the algebra on the invertible -module yields a -homomorphism of algebras

 \rm SymmT(A)→OT=EndT(m⋀T(A)).

This is a symmetric-tensor version of the norm map, usually given as a homogeneous polynomial; it can be written locally it terms of determinants. Applying Spec, we get a -map, called the canonical multisection of ,

 σZ/T:T→Z(m)T=\rm SpecT(\rm SymmT(A)).

This map is obviously compatible with base-change and satisfies a ’locality’ property, namely if with each flat of degree , ,then factors through

 ∏σZi/T:T→∏Z(mi)T.

Consequently, if and the fibre and each is supported at a unique point , then is the unique point of , usually denoted , and .

1.2. Cycle map

Let be a quasi-projective morphism, a morphism and a -valued point of the relative Hilbert scheme , i.e. a closed subscheme of that is finite flat of degree over . Examples of possible include the Hilbert scheme itself and any scheme mapping to it. We have the canonical multisection, which is a -morphism

 σZ/T:T→Z(m)T⊂(X×BT)(m)T=X(m)B×BT.

Composing with the projection, we get the cycle map, a -morphism

 cZ:T→X(m)B.

Again, this is compatible with base-change and has a locality property. Moreover, it depends only on quasi-intrinsically in the sense that if is any locally closed subscheme such that contains scheme-theoretically, then factors through . Also, there is an analogous and compatible construction in the analytic category.

2. Blowup Theorem: Set-up and preliminary reductions

2.1. Set-up

Let

 π:X→B

be a flat family of nodal, generically smooth curves with reduced and irreducible. Let , respectively, denote the th Cartesian and symmetric fibre products of relative to . Thus, there is a natural map

 ωm:XmB→X(m)B

which realizes its target as the quotient of its source under the permutation action of the symmetric group Let

 Hilbm(X/B)=X[m]B

denote the relative Hilbert scheme parametrizing length- subschemes of fibres of , and

 c=cm:X[m]B→X(m)B

the natural cycle map constructed above, associated to the universal subscheme . Let denote the discriminant locus or ’big diagonal’, consisting of cycles supported on points (endowed with the reduced scheme structure). Clearly, is a prime Weil divisor on , birational to (though it is less clear what the defining equations of on are near singular points). The main result of Sections 1-4 is the

Theorem 1 (Blowup Theorem).

The cycle map

 cm:X[m]B→X(m)B

is equivalent to the blowing up of .

The proof presented here is an elaboration of the one sketched in [13].

2.2. Reductions

We begin with some preliminary remarks and reductions. To begin with, recall that the cycle map is compatible with base-change, as was observed in §1, and note now that the same is true of the blowup of : indeed given a base-change , we have , hence also , so

 ⨁nInDm(XB′/B′)=(⨁nIDm)⊗OB′,

and applying Proj we get

 BℓDm(X(m)B)×BB′=BℓDm(XB′/B′)(XB′)(m)B′.

Because the Theorem is local over and locally any family is a base-change from a versal one, we may as well assume is a versal deformation of a nodal curve , and in particular and are smooth.

Next, the Theorem is the statement that the natural birational correspondence between and projects isomorphically both ways (in particular is irreducible). By GAGA, it suffices to prove for the corresponding analytic spaces. Then, since the statement is local over , we may work over a neighborhood of a given cycle , of the form where is a suitable analytic neighborhood of . The corresponding open subset of is just , where for an analytic open , is the set of schemes contained in . We note that this depends only on up to analytic isomorphism: e.g. because it can be identified with a Douady space of finite subspaces of ; or more directly, by GAGA, there is a natural correspondence between analytic families of finite subschemes contained in and finite analytic subspaces of . Now choosing appropriately, we may assume there is an open subset such that is a base-change of the family given by (the ’standard model’).

Now suppose we could show that , is the blowup of in . Then the same is true for To conclude that , it would suffice to show that

 BℓDm∏B(Ui)(mi)B≃∏B(BℓDm(Ui)(mi)B)

or equivalently,

 (2.1) BℓDm∏B(Ui)(mi)B≃∏∏B(Ui)(mi)B(BℓDm(Ui)(mi)B)

(2.1) holds because:

1. The local analysis of the next two sections will show, in particular, that is a small blowup, centered over the locus of schemes with multiplicity at the node, therefore so is .

2.  IDm(∐Ui/B)|∏B(Ui)(mi)B=∏IDmi(Ui/B).
3. The blowup centers are transverse for different .

4. The following general remark.

Remark 2.

Let be an arbitrary collection of ideals on a variety , not necessarily mutually transverse or even distinct.

(i) The blowup of the product ideal is the unique -dominating component of the fibre product . For simplicity we check this for . We may work locally over . If are generators for respectively, then the blowup of is covered by open affines whose coordinate rings are generated over by symbols satisfying the obvious relations , . Similarly with open affines for the blowup of , with generators are regular. There are obvious maps , defined by , leading to maps over

 BℓI1I2X⇄BℓI1X×XBℓI2X.

These clearly give an isomorphism as claimed.

Note that the foregoing argument makes no assumption regarding transversality of . In general, if are not transverse, e.g. , then is reducible: e.g. is a zero-divisor (usually nonzero) on . The dominating component of is .

(ii) In the above situation, if the are small blowups, i.e. for each the exceptional locus on (the center), i.e. the non-invertible locus of , is of codimension and its inverse image is of codimension , and if for different the centers are mutually transverse, then the fibre product is in fact irreducible, i.e. has no non-dominating components. This is because any non-dominating component would have to be of smaller dimension, whereas by semi-continuity, in the fibre product, which is the inverse image of the small diagonal in by the natural map

 ∏BℓIiX→Xk,

every component is of dimension .

We have now reduced the Theorem to the case where is the standard family , which we assume till further notice; we also let denote any neighborhood of the origin in .

3. A local model

We now give an explicit construction in coordinates of the relative Hilbert scheme of the standard family. This construction will have many applications beyond the proof of the Blowup Theorem. We begin with some preliminaries.

3.1. Symmetric product

Assuming has the local form , the relative Cartesian product , as a subscheme of , is given locally by

 x1y1=...=xmym=t.

Let denote the elementary symmetric functions in and in , respectively, where we set . We note that these functions satisfy the relations

 (3.1) σymσxj=tjσym−j,  σxmσyj=tjσxm−j, (3.2) tm−iσym−j=tm−i−jσxjσym,  tm−iσxm−j=tm−i−jσyjσxm

(of course the relations in second set follow from those of the first). Putting the sigma functions together with the projection to , we get a map

 σ:U(m)B=Symm(U/B)→A2mB=A2m×B
 σ=((−1)mσxm,...,−σx1,(−1)mσym,...,−σy1,π(m))

where is the structure map.

Lemma 1.

is an embedding locally near where is the origin in .

Proof.

It suffices to prove this formally, i.e. to show that generate the completion of the maximal ideal of in To this end it suffices to show that any -invariant polynomial in the is a polynomial in the and . Let us denote by the averaging or symmetrization operator with respect to the permutation action of , i.e.

 R(f)=1m!∑g∈Smg∗(f).

Then it suffices to show that the elements , where (resp. ) range over all monomials in (resp. ) are polynomials in the and . Because , we may assume are disjointly supported in the sense that . On the other hand, expanding the product we get a sum of monomials times a rational number; those with add up to , while those with not disjointly supported are divisible by . Thus,

 R(xIyJ)−m!R(xI)R(yJ)=tF

where is an -invariant polynomial in the of bidegree , hence a linear combination of elements of the form . By induction, is a polynomial in the and clearly so is Hence so is and we are done.∎

Remark 2.

It will follow from the Blowup Theorem 1 and its proof that the equations (3.1-3.2) actually define the image of scheme-theoretically (see Cor. 5 below); we won’t need this, however.

3.2. A projective family

Now we present a construction of our local model . This is motivated by our study in [15] of the relative Hilbert scheme of a node. As we saw there, the fibres of the cycle map are chains consisting of rational curves where takes the values from for the generic fibre (meaning the fibre is a singleton) to for the most special fibre. Therefore, it is reasonable to try to model the cycle map on a standard pencil of rational normal -tics specializing to a chain of lines. Further motivation for the construction that follows comes from [14], where an explicit construction is given for the full-flag Hilbert scheme.

Let be copies of , with homogenous coordinates on the -th copy. Let

 ~C⊂C1×...×Cm−1×B/B

be the subscheme over defined by

 (3.3) v1u2=tu1v2,...,vm−2um−1=tum−2vm−1.

This construction is motivated (cf. [14]) by viewing as a stand-in for where is of cardinality and etc; the ratio is independent of for fixed . That said, is in any event a reduced complete intersection of divisors of type

 (1,1,0,...,0),(0,1,1,0,...,0),...,(0,...,0,1,1)

(relatively over ) and it is easy to check that the fibre of over is

 (3.4) ~C0=m−1⋃i=1~Ci,

where

 ~Ci=[1,0]×...×[1,0]×Ci×[0,1]×...×[0,1]

and that in a neighborhood of the special fibre , is smooth and is its unique singular fibre over We may embed in relatively over using the plurihomogeneous monomials

 (3.5) Zi=u1⋯ui−1vi⋯vm−1,i=1,...,m.

These satisfy the relations

 (3.6) ZiZj=tj−i−1Zi+1Zj−1,i

so they embed as a family of rational normal curves specializing to , which is embedded as a nondegenerate, connected chain of lines.

3.3. To Hilb

Next consider an affine space with coordinates , . The are to play the roles of respectively (where as we recall plays that of ). With this and the relations (3.1), (3.2) in mind, let be the subscheme defined by

 (3.7) a0u1=tv1,d0vm−1=tum−1 a1u1=dm−1v1,...,am−1um−1=d1vm−1.

Note that comes equipped with a map to (via the projection to ), whence a projection

 pA2mB:~H→A2mB.

Set Then consider the subscheme of defined by the equations

 (3.8) (3.9) F1:=u1xm−1+u1am−1xm−2+...+u1a2x+u1a1+v1y∈Γ(Y,L1) ... Fi:=uixm−i+uiam−1xm−i−1+...+uiai+1x+uiai+vidm−i+1y+...+vidm−1yi−1+viyi (3.10) ∈Γ(Y,Li) ... (3.11) Fm:=d0+d1y1+...+dm−1ym−1+ym∈Γ(Y,OY).

The following statement essentially summarizes results from [15].

Theorem 3.
1. is smooth and irreducible.

2. The ideal sheaf generated by defines a subscheme of that is flat of length over and flat over .

3. The classifying map

 Φ=ΦI:~H→Hilbm(U/B)

is an isomorphism and via , the projection corresponds to the cycle map.

4. induces an isomorphism

 ~C0=(~C)0=p−1A2mB(0)→Hilb0m(X0)=m−1⋃i=1Cmi

(cf. [15]) of the fibre of over with the punctual Hilbert scheme of the node on the special fibre , in such a way that the point corresponds to

• the subscheme with ideal if

• the subscheme if ,

• the subscheme if

In particular, corresponds to .

5. over the standard open , a co-basis for the universal ideal (i.e. a basis for ) is given by

 1,...,xm−i,y,...,yi−1.
6. induces an isomorphism of the special fibre of over with , and is a divisor with global normal crossings where each is smooth, birational to , and for has special fibre under the cycle map .

Proof.

Assertions (i), (ii) are clear from the defining equations To prove (iii) and (iv) consider the point on the special fibre of over with coordinates

 vj=0, ∀j

Then has an affine neighborhood in defined by

 (3.12) Ui={uj=1, ∀j

and these cover a neighborhood of the special fibre of Now the generators admit the following relations:

 ui−1Fj=ujxi−1−jFi−1, 0≤ji

where we set for Hence is generated on by and assertions (iii), (iv) follow directly from Theorems 1,2 and 3 of [15] .

As for (v), it follows immediately from the definition of the , plus the fact just noted that, over the ideal is generated by , and that on , we can set Finally (vi) is contained in [15], Thm. 2.

At this point it’s worth noting the following consequences of Theorem 3, (i). First, recall that a deformation of a nodal curve is said to be locally versal (or locally versal at the nodes) if the natural map of to the product of local deformation spaces is smooth.

Corollary 4.

Let be a family of nodal or smooth curves.

(i) is a normal crossings morphism, i.e. fibres have normal crossings.

(ii) If is locally versal at the nodes, then and the universal subscheme over are smooth.

(iii) If is irreducible then so is

Remark.

In (ii), the smoothness claimed is of course in the absolute sense, i.e. over , not over .

Proof.

We first prove (ii) as (i) is similar and simpler. Working near a fibre , there is a standard coordinate neighborhood of each node , which is a pullback of , and such that the product map is smooth. Then is smooth over , and the latter is smooth. Therefore is smooth, hence so is .

(iii) It follows from the local models that the every fibre component of is dimensional and dominates a fibre component of . Since is irreducible, so is . ∎

In light of Theorem 3, we identify a neighborhood of the special fibre in with a neighborhood of the punctual Hilbert scheme (i.e. ) in , and note that the projection coincides generically, hence everywhere, with . Hence may be viewed as the subscheme of defined by the equations

 σxmu1=tv1, (3.13) σxm−1u1=σy1v1,...,σx1um−1=σym−1vm−1, tum−1=σymvm−1.

Alternatively, in terms of the coordinates, may be defined as the subscheme of defined by the relations (3.6), which define , together with

 (3.14) σyiZi=σxm−iZi+1, i=1,...,m−1.

4. Reverse engineering and proof of Blowup Theorem

Reverse-engineering an ideal means finding generators with given syzygies. Our task now is effectively to reverse-engineer an ideal (discriminant ideal) in the ’s whose syzygies, for suitable generators, are given by (3.14) and (3.6). This will be achieved by passing to the ordered version of Hilb, i.e. . The sought-for generators will be given by certain ’mixed Van der Monde’ determinants. The proof of the Blowup Theorem is then concluded, essentially by showing explicitly that, locally over Hilb, all the generators are multiples of one of them.

4.1. Order

Let , so we have a cartesian diagram

 OHm\lx@stackrelϖm⟶Hmocm↓□↓cmXmB\lx@stackrelωm⟶X(m)B

and its global analogue

 X⌈m⌉B\lx@stackrelϖm⟶X[m]Bocm↓□↓cmXmB\lx@stackrelωm⟶X(m)B

Here the horizontal maps are all -quotients, hence flat. Note that is normal and Cohen-Macaulay: this follows from the fact that it is a quotient by of , which is a locally complete intersection with singular locus of codimension (in fact, , since is smooth). Alternatively, normality of follows from the fact that is smooth and the fibres of are connected and reduced (being products of connected chains of rational curves), using the following general fact: if is a proper surjective morphism with connected reduced fibres between integral algebraic schemes over an algebraically closed field and is normal, then so is [proof: For any closed point , the inclusion is an isomorphism because is reduced and connected. By an easy composition series argument, the analogous statement holds with