On the rationality of the singularities of the $A_2$-loci

TL;DR

This paper investigates the singularities of the $A_2$-loci, demonstrating that they are generally worse than rational, which impacts their geometric and algebraic properties.

Contribution

It proves that the $A_2$-loci have singularities worse than rational, providing new insights into their geometric structure and limitations in their GIT quotient representations.

Findings

$A_2$-loci have non-rational singularities

Implications for GIT quotient constructions

Advances understanding of singularity types in algebraic geometry

Abstract

The rationality of the singularities of the $A_n$-loci is the natural question that arises in the papers devoted to the study of the Thom polynomials and $K$-theoretic invariants of the said loci. In this paper we prove that, in general, the $A_2$-loci have singularities worse than rational, and therefore they can not be presented as a GIT quotient of a smooth variety with respect to a reductive group.