Seminars on Introduction to Singularities

Lectures begin: Sep. 22, 2011
Exercises begin: TBA

Condition for Testal: Finish 1/3 of the exercises.

Exam: Evaluation of in-class presentation, and ability to answer questions about material presented; 20 min exam .

Exercises: Details of the exercises can be found here.

Abstract

Singularities arise in many mathematical contexts, and are both intrinsically beautiful and unavoidable in practice. However, in algebraic geometry, one of the main celebrated theorems of the 20th century is the so-called resolution of singularities in characteristic zero -- roughly, that singular spaces can be algorithmically replaced by smooth (non-singular) ones in a fashion that encodes all the details of the singularity. Traditionally considered an impenetrably difficult result, the proof in recent years has been reduced to what amounts to multi-variable calculus coupled to appropriate commutative algebra. We will focus on recent accessible expositions of the proof of this result, first in special cases (e.g., curves) and then in general.

Contents

Chapter 1, resolution for curves; Chapter 2, resolution for surfaces.

Main reference

• Kollar, Lectures on resolution of singularities. Annals of Mathematics Studies, 166. Princeton University Press, Princeton, NJ, 2007.