Algorithmic Resolution of Singularities

Abstract

Abstract Although the problem of the existence of a resolution of singularities in characteristic zero was already proved by Hironaka in the 1960s and although algorithmic proofs of it have been given independently by the groups of Bierstone and Milman and of Encinas and Villamayor in the early 1990s, the explicit construction of a resolution of singularities of a given variety is still a very complicated computational task. In this article, we would like to outline the algorithmic approach of Encinas and Villamayor and simultaneously discuss the practical problems connected to the task of implementing the algorithm. Introduction The problem of existence and construction of a resolution of singularities is one of the central tasks in algebraic geometry. In its shortest formulation it can be stated as: Given a variety X over a field K , a resolution of singularities of X is a proper birational morphism π : Y → X such that Y is a non-singular variety. Historically, a question of this type has first been considered in the second half of the 19th century – in the context of curves over the field of complex numbers. It was already a very active area of research at that time with a large number of contributions (of varying extent of rigor) and eventually lead to a proof of existence of resolution of singularities in this special situation at the end of the century.