2
$\begingroup$

Let $S^{[n]}$ be the Hilbert scheme of $n$ points on a smooth projective surface (actually, right now I am particularly interested in del Pezzo surfaces). Let $B$ be the exceptional divisor of the Hilbert-Chow morphism $S^{[n]} \rightarrow \operatorname{Sym}^n S$. Let $L$ be a divisor on $S$ and $\tilde L$ the corresponding "symmeterized" divisor on $S^{[n]}$, i.e. the set of $D \in S^{[n]}$ such that $\operatorname{Supp} D \cap \operatorname{Supp} L \neq 0$.

I would like to have a formula for $\chi(\tilde L+\frac{r}{2}B)$ in terms of $r$, $n$, and the invariants of $S$ and $L$. (We have a conjecture for del Pezzo surfaces by doing a different computation that we expect to match.)

In, e.g. [Li, Qin, and Wang], they say that there is an algorithm to compute the cup product of any two cohomology classes on $S^{[n]}$ for an arbitrary $S$. Furthermore, Boissiere has a paper about finding "universal formulas" for chern classes of tangent bundles of $S^{[n]}$.

If I understand the chern classes of the the tangent bundle, maybe I could understand the todd class, and assuming I could convert $\tilde L + \frac{r}{2}B$ into generators suitable for the algorithm mentioned by L,Q,W, maybe I could use GRR to compute the desired Euler characteristic.

I've been reading abstracts and introductions of these and surrounding papers, but I'm having a hard time getting a feel for what they can do. I don't mind putting in some time in to learn some new stuff, but I'd' like to be sure I'm going in the right direction first.

Note I previously asked this question for the special case $n=2$, and people pointed out in the comments that one could simply use a nice description of $S^{[2]}$ to do this computation, and indeed this turned out to be the case. But now I want to do this for arbitrary $n$.

Question Is the kind of stuff referenced above the right way to go? If so, do you have any recommendations for where to start reading? Or is there a different strategy that looks better?

$\endgroup$
2
  • 1
    $\begingroup$ The first few del Pezzos are toric, so $T^2$ acts on the corresponding Hilbert schemes, with isolated fixed points!, and your divisors can be chosen $T^2$-invariant. Then you can compute these Euler characteristics by equivariant localization, i.e. the Atiyah-Bott Riemann-Roch-Lefschetz Woods Hole theorem. For me, equivariantly is usually much easier than nonequivariantly. $\endgroup$ – Allen Knutson Feb 4 '15 at 6:58
  • $\begingroup$ @Allen: That would be sweet! I am excited to try this. $\endgroup$ – Drew Feb 4 '15 at 16:22
1
$\begingroup$

Here is the paper to look at.

http://arxiv.org/abs/math/9904095

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.