Definition of a Singularity, Resolutions of Singularities

Abstract

In this chapter we define singularities on analytic spaces or on algebraic varieties. Then, we introduce the fact that an isolated singularity on an analytic space can be regarded as a singularity on an algebraic variety. We also introduce Hironaka’s theorem stating that every algebraic variety over a field of characteristic zero has a resolution of the singularities. In this book our interest is focused on singularities. To this end, we study the resolved space instead of studying the singularity itself, therefore this resolution theorem is essential.