Ridge of the tangent cone of a singularity: a forgotten theorem
type: Item
instance of: scholarly article
title: Ridge of the tangent cone of a singularity: a forgotten theorem
zbMATH Open document ID: 1360.14011
author: Vincent Cossart
author: Olivier Piltant
author: Bernd Schober
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2017-04-26T00:00:00Z
review text: The authors give a proof of a theorem that, according to them, Jean Giraud ``forgot'' to state. The situation is the following one: let \(X\) be an excellent scheme and let \(H_X\) denote the modified Hilbert-Samuel function of \(X\). It is known that this function cannot increase under permissible blowing ups and that it is upper semi-continuous. The result proved in this paper is a strengthened version of this statement. More precisely, let us assume that \(X\) is embedded in some regular scheme. Then for every \(x\in X\) let us denote by \(\tau_{st}(x)\) the codimension of the ridge of the tangent cone of \(X\) at \(x\) (the ridge of the tangent cone is the largest additive group that preserves the tangent cone by translation). Then the function \[ \iota : X\longrightarrow \mathbb N^{\mathbb N}\times -\mathbb N\qquad \] \[ \qquad x\longmapsto (H_X(x),-\tau_{st}(x)) \] does not increase under permissible blowing ups for the lexicographic ordering and it is upper semi-continuous. When \(X\) is not assumed to be embedded in some regular scheme, the authors proves an analogue of this statement by replacing \(\tau_{st}\) by a modified version of this codimension.
reviewed by: Guillaume Rond
Mathematics Subject Classification ID: 14B05
Mathematics Subject Classification ID: 14B25
Mathematics Subject Classification ID: 14E15
zbMATH DE Number: 6708166
zbMATH Keywords: resolution of singularities
zbMATH Keywords: Hilbert-Samuel function
MaRDI profile type: Publication
full work available at URL: j.crma.2017.02.006
OpenAlex ID: W2554535384
cites work: On the characteristic functions of a local ring
cites work: Q4175355
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cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
cites work: Q5611879
DOI: 10.1016/J.CRMA.2017.02.006
description: scientific article; zbMATH DE number 6708166