The Betti numbers of the Hilbert scheme of points on a smooth projective surface
Let S be a smooth projective surface (over \({\mathbb{C}})\) and \(S^{[n]}\) the component of the Hilbert scheme of S which parametrizes subschemes of finite length n on S. Recently \textit{G. Ellingsrud} and \textit{S. A. Strømme} have computed the Betti numbers of \(S^{[n]}\) for \(S={\mathbb{P}}_ 2\) [Invent. Math. 87, 343-352 (1987; Zbl 0625.14002)]. In this paper the Betti numbers of \(S^{[n]}\) are computed for an arbitrary smooth projective surface S. For \(X=S\) or \(X=S^{[n]}\) let \(b_ i(X)\) be the i-th Betti number and \(p(X,z)=\sum_{i}b_ i(X)z^ i \) the Poincaré polynomial. Then the main result is: \[ \sum^{\infty}_{n=0}p(S^{[n]},z)t^ n=\prod^{\infty}_{k=1}\frac{(1+z^{2k-1}t^ k)^{b_ 1(S)}(1+z^{2k+1}t^ k)^{b_ 1(S)}}{(1-z^{2k-2}t^ k)^{b_ 0(S)}(1-z^{2k}t^ k)^{b_ 2(S)}(1-z^{2k+2}t^ k)^{b_ 0(S)}}. \] The proof uses reduction modulo q and the Weil conjectures. We also make some remarks about the Hodge numbers.
scientific article; zbMATH DE number 4112747
scientific article