Vanishing cycles of smoothable isolated Cohen-Macaulay codimension 2 singularities of type 2

TL;DR

This paper investigates the vanishing topology of smoothable isolated Cohen-Macaulay codimension 2 singularities, revealing a unique special vanishing cycle in dimensions 2 and 3 linked to their determinantal structure.

Contribution

It extends previous work to include possibly non-isolated singularities, identifying a unique vanishing cycle related to the determinantal structure in specific dimensions.

Findings

Exactly one special vanishing cycle in degree 2 in dimensions 2 and 3

The special cycle is closely related to the determinantal structure

Extension of previous results to non-isolated singularities

Abstract

We extend the results from the previous paper by A. Fr\"uhbis-Kr\"uger and the author [arXiv:1501.01915] to the vanishing topology of those singularities in the title. Studying the case of possibly non-isolated singularities in the Tjurina- transform, we reveal that in dimension 3 and 2 there always is exactly one special vanishing cycle in degree 2 closely related to the determinantal structure of the singularity.