Isosingular loci of algebraic varieties

TL;DR

This paper introduces the concept of isosingular loci for algebraic varieties, extending previous analytic results to algebraic cases, with special considerations for characteristic zero and positive characteristic scenarios.

Contribution

It defines isosingular loci for algebraic varieties and extends Ephraim's analytic results to algebraic varieties, addressing characteristic-dependent obstructions.

Findings

Partial extension of Ephraim's result in arbitrary characteristic

Full extension in characteristic zero

Identifies non-separability as an obstacle in positive characteristic

Abstract

We define the notion of isosingular loci of algebraic varieties, following the analytic case first studied by Ephraim. In particular, we give a partial extension of his main result in arbitrary characteristic and a full extension assuming characteristic $0$. One of the main obstructions in the positive characteristic case is the non-separability of the orbit map associated to the contact group, as first observed by Greuel and Pham for isolated singularities.