Resolution of Singularities and Birational Geometry

Purdue University, West Lafayette IN

Investigators

Abstract

The investigator plans to adapt his simplified algorithm of Hironaka's resolution of singularities to different types of resolution problems. The Hironaka theorem on resolution of singularities plays a central role in Algebraic Geometry. It constitutes a basis of proofs of many theorems in Algebraic and Analytic Geometry. The theorem on desingularization is proven in characteristic zero and is known only in some special cases in positive characteristic. This is a major drawback for many theorems which rely on desingularization and which can be proven in characteristic zero only. The project deals with studying some new ideas and developing a new relatively simple approach to the resolution problem in positive characteristic. By suitable modification of the basic notions (like order, marked ideals and etc) introduced in the algorithm in characteristic zero the investigator plans to approach the problem of resolution of singularities in positive characteristic. The new notions open some new possibilities in related areas. In particular, the new invariant proposed in the project, so called p-order, may have interesting applications in problems related to studying singularities in positive characteristic. The goal splits into two parts. First, the investigator plans to show how to reduce in a canonical way the resolution of general singularities to the singularities of the special form (singularities of Giraud hypersurfaces of maximal contact). Understanding these singularities is considered to be the key for solving the problem. Second, the investigator plans to approach the resolution of singularities by using the introduced language and the inductive scheme developed in the first part. Although the problem seems very difficult, using new tools significantly simplifies it and allows us to better understand the problem in many particular situations. Moreover the investigator plans to analyze and generalize the simplified algorithms of Hironaka desingularization and the Weak factorization from the point of view of further potential applications.