Determinantal Singularities

TL;DR

This paper surveys determinantal singularities, exploring their deformations, topology, and new phenomena such as non-isolated singularities and smoothings, extending classical results on complete intersections.

Contribution

It provides a comprehensive overview of determinantal singularities, highlighting their unique properties and recent developments in their deformation theory and classification.

Findings

Determinantal singularities include non-isolated and finitely determined cases.

Smoothings of these singularities can have low connectivity.

The survey covers classifications of simple determinantal singularities.

Abstract

We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena such as for instance non-isolated singularities which are finitely determined, or smoothings with low connectivity; already the union of the coordinate axes in $(\mathbb{C}^3,0)$ is determinantal, but not a complete intersection. We start with the algebraic background and then continue by discussing the subtle interplay of unfoldings and deformations in this setting, including a survey of the case of determinantal hypersurfaces, Cohen-Macaulay codimension $2$ and Gorenstein codimension $3$ singularities, and determinantal rational surface singularities. We conclude with a discussion of essential smoothings and provide an appendix listing known classifications of simple determinantal singularities.