Some results in the problem of simultaneous resolution of singularities

Nan Gu, Purdue University

Abstract

Given an algebraic function field K/k and a finite algebraic extension L of K, does there exist a projective variety X with function field K such that both X and its normalization Y in L are non-singular? If the answer is yes, we say that L/K is simultaneously resolvable, or that L/K admits a simultaneous resolution of singularities. In this monograph, we give an introduction to this problem and the techniques needed to handle the problem. We prove a theorem to calculate the integral closure of a Krull domain in a certain field extension, which generalizes a similar result due to Abhyankar. We then use it as the central algebraic tool to prove two new results for this problem, namely the simultaneous resolvability of n-fold cubic extensions and the existence of an extension with Galois group a direct sum of Zq and any finite group H that admits no simultaneous resolution of singularities.

Degree

Ph.D.

Advisors

Abhyankar, Purdue University.

Subject Area

Mathematics

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