Hilbert schemes of zero-dimensional subschemes of smooth varieties
type: Item
instance of: scholarly article
title: Hilbert schemes of zero-dimensional subschemes of smooth varieties
zbMATH Open document ID: 0814.14004
published in: Lecture Notes in Mathematics
publication date: 1994-04-24T00:00:00Z
review text: Let \(X\) be a smooth projective variety defined over an algebraically closed field \(k\). Let \(X^{[n]} := \text{Hilb}(X)\) be the Hilbert scheme of zero-dimensional subschemes of \(X\) of length \(n\). The main theme of the book is to study the Hilbert schemes \(X^{[n]}\), with particular reference to understand the cohomology and the Chow ring of \(X^{[n]}\). For \(n = 1,2\), the Hilbert scheme \(X^{[n]}\) is very well understood. In fact, \(X^{[1]}\) is just \(X\) itself and \(X^{[2]}\) can be obtained by blowing up \(X \times X\) along the diagonal and taking the quotient by the natural involution that exchanges factors in \(X \times X\). The Hilbert scheme \(X^{[n]}\) is smooth if \(n \leq 3\) or \(\dim X \leq 2\). If \(X\) is a curve, then \(X^{[n]}\) is isomorphic to the \(n\)-th symmetric product, \(X^{(n)}\), of \(X\). In general, in any dimension, the natural set- theoretic map \(X^{[n]} \to X^{(n)}\) sending each 0-scheme to its support (with multiplicities) gives a desingularization of \(X^{(n)}\) if \(X^{[n]}\) is smooth. When \(\dim X = 2\), the canonical bundle of \(X^{[n]}\) is the pullback of the dualizing sheaf of \(X^{(n)}\). In chapter 1 some fundamental facts and background material are recalled. In \S1.1 the definition and the most important properties of \(X^{[n]}\) are given. In \S1.2 the author discusses the Weil conjectures in the form he needs to compute Betti numbers of Hilbert schemes. In \S1.3 the punctual Hilbert scheme, which parametrizes subschemes concentrated in a point of a smooth variety is studied. Chapter 2 is devoted to the computation of the Betti numbers of Hilbert schemes. In \S\S2.1, 2.2 the structure of the closed subscheme of \(X^{[n]}\) which parametrizes subschemes of length \(n\) on \(X\) concentrated on a variable point of \(X\), and the punctual Hilbert schemes \(\text{Hilb}(kx,y)\) are studied. \S2.3 contains the explicit computation of the Betti numbers of \(S^{[n]}\) for a smooth surface \(S\), using the Wei…
reviewed by: Mauro C. Beltrametti
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14F45
Mathematics Subject Classification ID: 14C25
Mathematics Subject Classification ID: 14-02
Mathematics Subject Classification ID: 14-01
zbMATH DE Number: 556121
zbMATH Keywords: Bibliography
zbMATH Keywords: punctual Hilbert scheme
zbMATH Keywords: Betti numbers
zbMATH Keywords: higher order Kummer varieties
zbMATH Keywords: triangle varieties
zbMATH Keywords: Chow ring
author: Lothar Göttsche
MaRDI profile type: Publication
Wikidata QID: Q30054474
full work available at URL: https://doi.org/10.1007/bfb0073491
OpenAlex ID: W579869728
DOI: 10.1007/BFB0073491
description: scientific article; zbMATH DE number 556121