Two considerations prompt us to look for algebraic approaches to the standard conjecture. The first is simply our pride as algebraic geome- ters. The second is that Harer’s approach to the calculation ofHi(Mg) becomes much harder to carry out with each increase ini; already with i=4 we appear to be reaching the limits of human patience and per- severance (although Harer and some of his students have done work on the next cases). It seems unlikely that his methods can be pushed much further.
The most promising strategy is to try to solve the problem in two steps: first, understand the cohomology of some parameter space or
spaces; second, by studying the maps from these spaces to moduli, deduce the cohomology ofMg. For example, the irreducibility ofMg
(which amounts to the calculation ofH0(Mg)) is proved by showing that the Hurwitz schemes are irreducible and then showing that every Mg is dominated by such a scheme.
Here we simply want to mention some conjectures on the cohomol- ogy of the Hilbert scheme that are in close analogy to the standard conjecture above. We may summarize the approach taken above to generating the stable cohomology of the moduli space of curves as follows. First, look for a canonically defined line bundle on the uni- versal curveCg, or, roughly equivalently at least up to torsion, for a consistently defined line bundle on each fiber of Cg. Then, take the Chern class of this bundle, raise it to various powers and take their Gysin images in the cohomology ofMg.
We may do exactly the same thing in the case of Hilbert schemes H = Hd,g,r. The main difference is that there are nowtwocanonically defined bundles on the universal curveπ :X✲H: the relative dual- izing sheafω = ωX/H as before, and the line bundle OX(1)pulled back fromOPr(1)by the inclusion ofXinPr× H. We may thus take all monomials in the Chern classes η and ξ of these two line bun- dles, and push them forward. We ask: do these Gysin images generate the Chow or homology ring of the restricted Hilbert schemeRin low codimension? For example, we might expect that the groupA1(R)of codimension 1 cycle classes in R will have rank 3, being generated overQby the classes
A=π∗(ξ2), B=π∗(ξ·η) and C=π∗(η2).
We may call this statement the standard conjecture for the low- dimensional cohomology of the Hilbert scheme.
We may make an analogous conjecture on the low-dimensional co- homology of the universal curveXR✲R: that it’s generated by the classes η and ξ over the ring H∗(R). In particular, this would say that the relative Picard group ofXR/R(the group of line bundles on XR modulo those pulled back from R) is generated by η andξ. In the formulation “ifLC is a rationally determined line bundle on every curveC in an open set of the Hilbert scheme, then for somen and m,LC ω⊗nC ⊗ OC(m)”, this conjecture is already in Enriques. Just as forMgandMg these conjectures lead to statements that the low- codimensional cohomology ofRis generated by standard classes and boundary components which we won’t formulate here.
Unfortunately, these conjectures arefalsein general. There are ex- amples of components of restricted Hilbert schemesRsuch that the universal curve XR✲R admits many line bundles (modulo those pulled back from R) other than ωX/B and OX(1). The simplest of these is given in the two exercises below.
D. Geometric and topological properties 65 Exercise (2.44) Let S0 be a smooth cubic surface andL1, . . . , L6 be disjoint lines onS0. LetC0⊂S0be a general curve in the linear system
C0∈ |nH−L1−2L2− · · · −6L6|.
Show that ifnis sufficiently large, then 1) C0is smooth and irreducible;
2) a general curve C in a componentH of the Hilbert scheme con- taining[C0]also lies on a smooth cubic surfaceS; and,
3) the class of the curveC⊂Sis expressible in the formnH− i·Li
for auniquechoice of 6 skew linesL1, . . . , L6⊂S.
How large doesnhave to be for each of these assertions to hold?
Exercise (2.45) LetX✲H be the universal curve over the compo- nentHof the Hilbert scheme described in the preceding exercise and XR✲Rits restriction to the open set of smooth curves. Show that the classes H andL1, . . . , L6 give rise to seven independent line bundles onXR, whose restrictions to a general fiberC⊂ XRare independent.
In other words, show that the group of rationally determined line bun- dles on XR✲Rhas rank at least 7. For extra credit, show that the rank is exactly 7.
Exercise (2.46) Continuing our analysis of the Hilbert scheme de- scribed in the preceding exercise, consider the Gysin images of the pairwise products of the classes H and L1, . . . , L6. Show that these give rise to at least four independent divisor classes on H, thus vi- olating the standard conjectures on the Picard group of the Hilbert scheme.
It’s therefore somewhat remarkable that for r = 1 and 2 (that is, for the Hurwitz scheme and Severi variety),the standard conjectures do seem to hold. Why this should be is unclear. The situation is com- pletely analogous with those considered in Chapter 1: the Hurwitz scheme and Severi variety are always irreducible of the correct di- mension, while the Hilbert scheme is in general neither. The basic references for the Severi variety case of these conjectures are [33], [32] and [36]: in the last it’s shown that a verification of the standard conjecture on the Picard group of Severi varieties would imply Harer’s theorem on the Picard group ofMg. Diaz and Edidin [31] have some results in the Hurwitz scheme case.
There is a further point to be made about the standard conjectures for Hilbert schemes of curves in higher-dimensional space. This is that, empirically, the components of the Hilbert scheme that violate the conjectures all lie over relatively small subvarieties ofMg— ones of codimension on the order ofgor more. This phenomenon has been
sufficiently often observed that we may include it in the statement of the conjectures. We include these here (for the Picard groups of the Hilbert schemes and their universal curves) for reference. We start with the Enriques conjecture:
Conjecture (2.47) (Enriques conjecture) 1) Let Hd,g be the space of branched covers π : C✲P1 of P1 of degree d and genus g. IfLC is a rationally determined line bundle on every curveC in an open set ofHd,g, then for somenandm,LC ω⊗nC ⊗π∗OP1(m).
2) Similarly, let Vd,g be the locus of reduced and irreducible plane curves of degreedand genus g. IfLC is a rationally determined line bundle on every curveC in an open set ofVd,g, then for somenand m,LC ω⊗nC ⊗ OC(m).
3) There exist real numbers α = α(g) > 0and β = β(g)such that the following statement holds: ifH is any component of the restricted Hilbert scheme of curves of degreedand genusginPr such that the induced rational map ϕ : H ✲Mg has image of codimension less than or equal to α·g+β, and if LC is a rationally determined line bundle on every curveC in an open set ofVd,g, then for somenand m,LC ω⊗nC ⊗ OC(m).
Exercise (2.48) Show that any one of these three statements implies the Enriques-Franchetta conjecture. (The converse, in casedis suffi- ciently large with respect tog, was established by Ciliberto [25].)
The statement of the standard conjecture for the low-dimensional cohomology of the Hilbert scheme (again just for the Picard group) is slightly more delicate, since it matters just what open subset of the Hurwitz/Severi/Hilbert scheme we choose. Probably the cleanest statement in the case of the Hurwitz and Severi schemes involves the smallest open subset: in the case of the Hurwitz scheme, the variety Hd,g of branched covers with simple branching, and, in the case of the Severi variety, the open setUd,g ⊂Vd,gof irreducible plane curves of degreedand geometric genusgwith only nodes as singularities. In both cases, we expect first that the classesA,BandC will be torsion and second that the rank of the Picard group is 0. For the Severi variety, the first expectation is proved in [33]. Of course, this would give the second if we knew that these classes also generated the Picard group.
Diaz and Harris actually show that the converse also holds: if the rank is 0, then these classes must actually generate Pic(Vd,g)⊗Q.
Conjecture (2.49) (Standard conjecture for Picard groups of parameter spaces)
1) Pic(Hd,g)⊗Q=0;
2) Pic(Ud,g)⊗Q=0; and
D. Geometric and topological properties 67 3) There exist real numbers α = α(g) > 0and β = β(g)such that the following statement holds: ifH is any component of the restricted Hilbert scheme of curves of degreedand genusgin Pr andH ⊂ H the open subset parameterizing smooth curves, such that the induced map ϕ : H✲Mg has image of codimension less than or equal to α·g+β, then the Picard groupPic(H)⊗QofH is generated by the classesA,BandC.
You may feel that the formulation of the third part of each of these conjectures in terms of unspecified constants α(g) and β(g) is a cheat: the statement as a result is so vague as to be virtually immune to counterexample. We agree. The problem is, the evidence available doesn’t give a clear indication of what the correct values of these con- stants should be. For the examples of which we know,α=1 andβ=0 should work; but that may not be the strongest possible statement.
We leave it instead as a challenge:
Problem (2.50) Can you find a componentHof the Hilbert scheme whose image in Mg has codimension less than g that violates the statement of either conjecture above?
Exercise (2.51) Calculate the codimension inMgof the image of the component of the Hilbert scheme introduced in Exercise (2.44); in par- ticular, observe that it’s greater thang.
Finally, we should say that there are analogous conjectures about the dimension and irreducibility of the Hilbert scheme. As we re- marked, while the Hurwitz and Severi varieties are always irreducible of the expected dimension, neither is true of the Hilbert schemeHd,g,r
in general. But, it may be conjectured that, as in the two conjec- tures above, the corresponding statementsdohold for components of Hd,g,r whose images inMghave relatively small codimension. We will discuss this briefly following the proof of the Brill-Noether theorem in Chapter 5.