Local factorization of birational maps
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.
scientific article; zbMATH DE number 1117744