On Tjurina Transform and Resolution of Determinantal Singularities

TL;DR

This paper investigates the Tjurina transform associated with determinantal singularities, exploring its properties and limitations in resolving these complex algebraic structures.

Contribution

It provides a detailed analysis of the Tjurina transform's properties and discusses its effectiveness and limitations in resolving determinantal singularities.

Findings

Tjurina transform has specific properties relevant to determinantal singularities.

The transform can sometimes be used to find resolutions, but not always.

Insights into the algebraic structure of determinantal singularities.

Abstract

Determinantal singularities are an important class of singularities, generalizing complete intersections, which recently have seen a large amount of interest. They are defined as preimage of $M^{t}_{m,n}$ the sets of matrices of rank less than $t$. The linear algebraic structure $M^{t}_{m,n}$ gives rise to some interesting structures on determinantal singularities. In this article we will focus on one of these, namely the Tjurina transform. We will show some properties of it, and discuss how it can and how can not be used to find resolutions of determinantal singularities.