A note on the Hilbert scheme of points on a cusp curve
type: Item
instance of: scholarly article
zbMATH Open document ID: 1244.14005
DOI: 10.1142/S0218196712500257
author: Hwayoung Lee
published in: International Journal of Algebra and Computation
publication date: 2012-06-04T00:00:00Z
Mathematics Subject Classification ID: 14C05
zbMATH DE Number: 6042661
zbMATH Keywords: Hilbert schemes
zbMATH Keywords: cusp curve
zbMATH Keywords: punctual Hilbert schemes
MaRDI profile type: Publication
cites work: A note on Hilbert schemes of nodal curves
cites work: Deformations of Algebraic Schemes
full work available at URL: s0218196712500257
OpenAlex ID: W2082823752
title: A note on the Hilbert scheme of points on a cusp curve
review text: The author investigates the Hilbert scheme of points on a cusp curve \(X\). Specifically, let \(X\) be a projective cusp curve with a cusp locally defined by \(x^2=y^3\) and let \(H\) denote the Hilbert scheme parameterizing zero-dimensional length \(m\) subschemes \(X\), for a fixed \(m\). Also let \(H^0\) denote the subscheme of \(H\) parameterizing those subschemes supported only at the cusp. The author establishes the two nice theorems:NEWLINENEWLINE1. The reduced scheme \((H^0)^{\mathrm{red}}\) is isomorphic to \(\mathbb{P}^1\) for \(m \geq 2\).NEWLINENEWLINE2. \(H\) has one singular point along \(H^0\).NEWLINENEWLINEThe proofs are explicit and thorough and will be of interest to those studying Hilbert schemes and in particular punctual Hilbert schemes.
reviewed by: Morgan Sherman
description: scientific article; zbMATH DE number 6042661