Jet schemes of complex plane branches and equisingularity
type: Item
instance of: scholarly article
title: Jet schemes of complex plane branches and equisingularity
zbMATH Open document ID: 1247.14011
author: Hussein Mourtada
publication date: 2012-06-06T00:00:00Z
full work available at URL: https://arxiv.org/abs/1009.5845
full work available at URL: https://eudml.org/doc/219717
reviewed by: Caterina Cumino
Mathematics Subject Classification ID: 14E18
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 6043080
zbMATH Keywords: jet schemes
zbMATH Keywords: singularities of plane curves
MaRDI profile type: Publication
OpenAlex ID: W2129077395
arXiv ID: 1009.5845
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cites work: Studies in Equisingularity I Equivalent Singularities of Plane Algebroid Curves
cites work: Q4078152
DOI: 10.5802/AIF.2675
published in: Annales de l’institut Fourier
review text: The \(m^{th}\)-jet scheme \(X_m\) of an algebraic variety \(X\) over an algebraically closed field \(k\) is a \(k\)-scheme of finite type which parametrizes morphisms \(\mathrm{Spec}(k[t]/t^{m+1})\to X\). Jet schemes have attracted, from various viewpoints, the attention of many authors such as \textit{J. F. Nash, jun.} [Duke Math. J. 81, No.1, 31--38 (1995; Zbl 0880.14010)] and, more recently, \textit{M. Mustată} [Invent. Math. 145, No. 3, 397--424 (2001; Zbl 1091.14004)], with \textit{L. Ein} and \textit{R. Lazarsfeld} [Compos. Math. 140, No. 5, 1229--1244 (2004; Zbl 1060.14004)].NEWLINENEWLINEIn the present paper the author considers a curve \(C\) in the complex plane, with a singularity at \(O\) at which it is analytically irreducible (i.e. the formal neighborhood \((C,O)\) of \(C\) at \(O\) is a branch). He determines the irreducible components of the \(m\)-Jet scheme of \((C,O)\) and shows that their number is not bounded as \(m\) grows. He also gives formulas for their number and for their codimension, in terms of \(m\) and of the generators of the semigroup of \((C,O)\).NEWLINENEWLINEAs it is well known, the semigroup of the branch \((C,O)\) and the topological type of \(C\) near \(O\) are equivalent data and characterize the equisingularity class of \((C,O)\), as defined by Zariski. The author shows in particular that the structure of the Jet Scheme determines the topological type of \(C\) and conversely.
description: scientific article; zbMATH DE number 6043080