Hurwitz Stacks of Groups Extensions and Irreducibility

TL;DR

This paper investigates the structure of special loci of algebraic curves with symmetry groups formed as extensions, using mixed étale cohomology to identify irreducible components and propose a heuristic for constructing further loci.

Contribution

It introduces relative Hurwitz data and applies mixed étale cohomology to classify irreducible components of certain symmetry loci in algebraic curves.

Findings

Identifies irreducible components of special loci using cohomology

Provides examples supporting the heuristic for building loci

Connects group extensions with geometric properties of curves

Abstract

We study the irreducible components of special loci of curves whose group of symmetries is given as certain group extension. We introduce some relative Hurwitz data, which we show by using mixed \'etale cohomology theory, identifies some irreducible components for rational and normal non-abelian special loci and Hurwitz spaces. A heuristic, that is supported by three classes of examples, provides an additional context for building further irreducible loci.