On proximity relations for valuations dominating a two-dimensional regular local ring
type: Item
instance of: scholarly article
title: On proximity relations for valuations dominating a two-dimensional regular local ring
zbMATH Open document ID: 0968.13011
DOI: 10.4171/RMI/267
author: José J. Aparicio
author: Ángel Granja
author: T. Sánchez-Giralda
published in: Revista Matemática Iberoamericana
publication date: 2000-06-29T00:00:00Z
full work available at URL: https://eudml.org/doc/39576
review text: Singularities of plane curves can be studied using quadratic transformations of regular local rings of dimension 2. This is a classical topic in commutative algebra and algebraic geometry. Iterated quadratic transformations applied to a given regular local ring of dimension 2 yield an infinite sequence of regular local rings, \(R_0\subset R_1\subset R_2\dots\), whose union is a valuation ring. The relationship between the rings in the sequence can be described by the so-called proximity relation. The authors introduce a sequence of rational numbers as a new numerical invariant of the proximity relation and call it the proximity sequence. It is shown that the proximity sequence encodes the same information as the refined proximity matrix. The sequences that arise as proximity sequences are characterized, and their connection with curve singularities is discussed.
reviewed by: Niels Schwartz
Mathematics Subject Classification ID: 13H05
Mathematics Subject Classification ID: 14H20
Mathematics Subject Classification ID: 13F30
zbMATH DE Number: 1419790
zbMATH Keywords: quadratic transformations of regular local rings
zbMATH Keywords: valuation ring
zbMATH Keywords: proximity relation
zbMATH Keywords: proximity sequence
zbMATH Keywords: curve singularities
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1419790