On the Vanishing Topology of Isolated Cohen-Macaulay Codimension 2 Singularities

TL;DR

This paper investigates the vanishing topology of isolated Cohen-Macaulay codimension 2 singularities, explicitly computes Betti numbers, and explains new phenomena using Tjurina modifications, revealing deeper insights into their topological structure.

Contribution

It provides explicit Betti number calculations and explains novel topological phenomena in Cohen-Macaulay codimension 2 singularities using Tjurina modifications.

Findings

Explicit Betti numbers for the singularities are determined.

New topological phenomena are identified and explained.

Tjurina modifications relate these singularities to complete intersections.

Abstract

Isolated Cohen-Macaulay codimension 2 singularities share many common features with isolated complete intersection singularities, but they also exhibit some striking new behaviour. One such instance was recently observed by Damon and Pike in their study of the vanishing topology and Euler characteristic, where they took this class of singularities as examples. In this article, we explore their findings further by determining the Betti numbers explicitly and explain the new phenomena. An important tool here is the Tjurina modification relating a Cohen-Macaulay codimension 2 singularity to a finite number of complete intersection singularities.