Equisingularity classes of birational projections of normal singularities to a plane

TL;DR

This paper classifies all equisingularity types of complete ideals in a two-dimensional regular local ring whose blow-ups have a specified normal surface singularity, solving a key discrete classification problem in singularity theory.

Contribution

It provides a complete solution to the discrete classification of equisingularity types for certain normal surface singularities, advancing the understanding of their birational projections.

Findings

Complete classification of equisingularity types for the discrete part.

Connection established between equisingularity classes and the moduli of plane curve singularities.

Resolution of a longstanding problem posed by Spivakovsky in 1990.

Abstract

Given a birational normal extension S of a two-dimensional local regular ring R, we describe all the equisingularity types of the complete ideals J in R whose blowing-up has some point at which the local ring is analytically isomorphic to S. The problem of classifying the germs of such normal surface singularities was already posed by Spivakovsky (Ann. of Math. 1990). This problem has two parts: discrete and continous. The continous part is to some extent equivalent to the problem of the moduli of plane curve singularities, while the main result of this paper solves completely the discrete part.