Local factorization of birational maps
type: Item
instance of: scholarly article
title: Local factorization of birational maps
zbMATH Open document ID: 0934.14006
author: Steven Dale Cutkosky
published in: Advances in Mathematics
publication date: 1998-03-16T00:00:00Z
review text: Inclusions of regular local rings \(R\subset S\) of dimension two with common quotient field have (according to a well known theorem of Zariski-Abhyankar) a simple structure, namely: \(R\subset S\) can be factored by a unique finite product of quadratic transforms. In dimension \(\geq 3\) the situation is a lot more complicated. In this sense Abhyankar made the following conjecture [see \textit{S. S. Abhyankar}, ``Ramification theoretic methods in algebraic geometry'', Ann. Math. Stud. 43 (1959; Zbl 0101.38201)]: Assume that \(K\) is a field of algebraic functions over a field \(k\), and \(R\) and \(S\) are regular local rings, essentially of finite type over \(k\), with quotient field \(K\). Let \(V\) be a valuation ring which dominates \(R\) and \(S\). Then there exists a regular local ring \(T\), essentially of finite type over \(k\), with quotient field \(K\), dominated by \(V\), containing \(R\) and \(S\), such \(R\subset T\) and \(S\subset T\) can be factored by products of monoidal transforms. The aim of the present paper under review is to prove a fundamental local theorem that implies Abhyankar's conjecture in dimension \(3\). Using his result, the author also proves the following global result (which partially answers a question of Hironaka and Abhyankar): Let \(k\) be a field of characteristic zero, \(\varphi:X \to Y\) a birational morphism of integral nonsingular proper excellent \(k\)-schemes of dimension \(3\). Then there exists a nonsingular proper \(k\)-scheme \(Z\) and birational morphisms \(f:Z\to X\) and \(g:Z\to Y\) such that \(\varphi \circ f=g\), with \(f\) and \(g\) locally products of monoidal transforms.
Mathematics Subject Classification ID: 14E05
Mathematics Subject Classification ID: 13F30
Mathematics Subject Classification ID: 13B02
zbMATH DE Number: 1117744
zbMATH Keywords: regular local rings
zbMATH Keywords: valuation ring
zbMATH Keywords: quadratic transforms
zbMATH Keywords: monoidal transforms
zbMATH Keywords: birational morphism
reviewed by: Lucian Bădescu
MaRDI profile type: Publication
full work available at URL: aima.1997.1675
OpenAlex ID: W2004068181
DOI: 10.1006/AIMA.1997.1675
description: scientific article; zbMATH DE number 1117744