Arc structure of singularities
type: Item
instance of: scholarly article
title: Arc structure of singularities
zbMATH Open document ID: 0880.14010
DOI: 10.1215/S0012-7094-95-08103-4
author: John F. jun. Nash
published in: Duke Mathematical Journal
publication date: 1996-07-07T00:00:00Z
review text: The author introduces a new approach in the study of singularities of algebraic varieties, the investigation of its ``arc structure''. In his terminology, an arc on a variety \(V\) defined over the complex numbers, is a germ of a parametrized algebroid curve on the variety, i.e. a morphism from \((T,0)= \text{Spec} \mathbb{C} t\) into \(V\), given by convergent power series. Finiteness properties of the set of arcs on \(V\), which do not factor through the singular locus, and send 0 into a given algebraic subvariety \(W\) of \(V\), are derived from the existence of a resolution of singularities of \(V\). A decomposition of this set into a finite number of families, depending only on the pair \((V,W)\), follows naturally; moreover for any resolution \(X \) of \(V\), the families are shown to be in \(1-1\) correspondence with a subset of the irreducible components of the inverse image of \(W\) in \(X\). The components so distinguished are essential components with respect to \(W\), appearing up to birational equivalence in all resolutions. A natural question arises; does every essential component come from an arc family? For surfaces, this amounts to decide whether there are as many families of arcs associated with \(W\) as irreducible components of its inverse image in the minimal resolution. A few examples are presented at the end. The paper has been written in the midsixties. \textit{D. Mumford} mentions its influence in the introduction of the book by \textit{G. Kempf}, \textit{F. Knudsen}, \textit{D. Mumford} and \textit{B. Saint-Donat}, ``Toroidal embeddings'' I, Lect. Notes Math. 339 (1973; Zbl 0271.14017)]. The problems stated above remain unsolved at the present time. Reference to further related development are given in ``Sur l'espace des courbes tracées sur une singularité'' by \textit{G. Gonzalez-Sprinberg} and \textit{M. Lejeune-Jalabert} [in: Algebraic Geometry and Singularities, Proc. 3rd internat. Conf., La Rabida 1991, Prog. Math. 134, 9-32 (1996;…
Mathematics Subject Classification ID: 14E15
Mathematics Subject Classification ID: 14H20
zbMATH DE Number: 880067
zbMATH Keywords: resolution of singularities
zbMATH Keywords: arc structure
zbMATH Keywords: singularities
zbMATH Keywords: essential components
Wikidata QID: Q110828745
MaRDI profile type: Publication
cites work: Q3260091
cites work: Resolution of singularities of an algebraic variety over a field of characteristic zero. I
full work available at URL: s0012-7094-95-08103-4
OpenAlex ID: W2034949131
description: scientific article; zbMATH DE number 880067