The Betti numbers of the Hilbert scheme of points on a smooth projective surface
type: Item
instance of: scholarly article
title: The Betti numbers of the Hilbert scheme of points on a smooth projective surface
zbMATH Open document ID: 0679.14007
DOI: 10.1007/BF01453572
published in: Mathematische Annalen
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164634
review text: 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.
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 4112747
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: Betti numbers
zbMATH Keywords: Poincaré polynomial
zbMATH Keywords: Weil conjectures
zbMATH Keywords: Hodge numbers
author: Lothar Göttsche
describes a project that uses: goettsche.lib
MaRDI profile type: Publication
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OpenAlex ID: W1538893142
description: scientific article; zbMATH DE number 4112747