Murphy's law in algebraic geometry: Badly-behaved deformation spaces
type: Item
instance of: scholarly article
title: Murphy's law in algebraic geometry: Badly-behaved deformation spaces
zbMATH Open document ID: 1095.14006
author: R. Vakil
published in: Inventiones Mathematicae
publication date: 2006-06-16T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0411469
review text: The author considers the following question: ``How bad can the deformation space of an object be?'' It is shown that usually the deformation space may be as bad as possible according to the philosophy ``There is no geometric possibility so horrible that it cannot be found generically on some component of some Hilbert scheme'' [cf. \textit{J. Harris, I. Morrison}, ``Moduli of Curves'', Grad. Texts Math. 187, Springer (1998; Zbl 0913.14005)]. By definition Murphy's law holds for a moduli space if every singularity type of finite type over \(\mathbb Z\) appears on that moduli space. It is proved that a lot of the well understood moduli spaces satisfy Murphy's law. For instance the Hilbert scheme of nonsingular curves in projective space, the Hilbert scheme of surfaces in \(\mathbb P^4\), the versal deformation spaces of isolated normal Cohen--Macaulay threefold singularities.
reviewed by: Gerhard Pfister
Mathematics Subject Classification ID: 14B12
Mathematics Subject Classification ID: 14C05
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14H50
Mathematics Subject Classification ID: 14B07
Mathematics Subject Classification ID: 14D22
Mathematics Subject Classification ID: 14B05
zbMATH DE Number: 5033665
zbMATH Keywords: deformation
zbMATH Keywords: moduli space
zbMATH Keywords: Murphy's law
MaRDI profile type: Publication
OpenAlex ID: W1994327923
arXiv ID: math/0411469
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DOI: 10.1007/S00222-005-0481-9
description: scientific article