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]](http://arxiv.org/pdf/math/0107139v2.pdf), 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](http://arxiv.org/pdf/math/0410458v2.pdf) 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](http://arxiv.org/pdf/math/0107139v2.pdf), 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 $n=2$](https://mathoverflow.net/questions/185229/computing-euler-charactistics-of-line-bundles-on-hilbert-schemes-of-points-on-su), 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?



Computing Euler Characteristics of Line Bundles on the Hilbert Scheme of n points
Here is the paper to look at.


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


