Hilbert scheme of points on cyclic quotient singularities of type {\((p,1)\)}
type: Item
instance of: scholarly article
title: Hilbert scheme of points on cyclic quotient singularities of type {\((p,1)\)}
zbMATH Open document ID: 1389.14003
author: Ádám Gyenge
published in: Periodica Mathematica Hungarica
publication date: 2017-11-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1603.02114
full work available at URL: 44389.pdf
review text: Let \(X\) be a complex quasi-projective surface and let \(n\) be a positive integer. The Hilbert scheme \(\hbox{Hilb}^n(X)\) parametrizes the zero-dimensional subschemes of \(X\) of length \(n\). The author studies the topological Euler characteristics of these Hilbert schemes, which he collects in a generating series \[ Z_X(q) = \sum_{n \geq 0} q^n \chi(\hbox{Hilb}^n(X)). \] When \(X\) is a smooth surface, these have been carefully studied by Fogarty, and Göttsche described the generating series of the Poincaré polynomial of these Hilbert schemes in terms of the Betti numbers of \(X\). To extend this, the author considers a particular action of \(\mathbb Z_p\) (\(p\) a positive integer) on \(\mathbb C^2\) involving another integer \(q\) coprime to \(p\), and denotes by \(X(p,q)\) the quotient variety. The main result of this paper is a representation of \(Z_{X(p,1)}(q)\) as a coefficient of a two-variable generating function, obtained by studying a torus action on \(X(p,1)\). The author links the combinatorics of this problem to \(p\)-fountains, a generalization of the notion of a fountain of coins.
reviewed by: Juan C. Migliore
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 6806101
zbMATH Keywords: Hilbert scheme
zbMATH Keywords: cyclic quotient singularity
zbMATH Keywords: \(p\)-fountain
MaRDI profile type: Publication
OpenAlex ID: W2964299586
arXiv ID: 1603.02114
cites work: Q4218535
cites work: The Betti numbers of the Hilbert scheme of points on a smooth projective surface
cites work: Q3836535
cites work: The Editor's Corner: n Coins in a Fountain
DOI: 10.1007/S10998-016-0146-Z
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description: scientific article