# Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points

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 $S^{}$ 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?

• 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. – Allen Knutson Feb 4 '15 at 6:58
• @Allen: That would be sweet! I am excited to try this. – Drew Feb 4 '15 at 16:22