The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
How can I compute the top interesection of $2n$ line bundles given in this basis on $S^{[n]}$? My particular problem needs to compute $(L_n + E)^{2n}$.
There is a lot of good stuff that can be computed for Hilbert schemes of surfaces, so I hope that there is a nice answer in terms of the invariants of $L$ and $S$. I've looked at a few papers and haven't found it, but it seems like something that has probably been done.
