Genus stabilization for moduli of curves with symmetries

TL;DR

This paper proves that for large genus quotient curves, the moduli space of algebraic curves with a fixed homological invariant under a finite group action is irreducible, linking stable classes to admissible invariants.

Contribution

It establishes irreducibility of moduli spaces with fixed homological invariants for large genus curves, extending previous unramified case results.

Findings

Moduli space is irreducible for large genus g'

Non-empty iff the class is admissible

Stable classes correspond to admissible classes

Abstract

In a previous paper, arXiv:1206.5498, we introduced a new homological invariant $\e$ for the faithful action of a finite group G on an algebraic curve. We show here that the moduli space of curves admitting a faithful action of a finite group G with a fixed homological invariant $\e$, if the genus g' of the quotient curve is sufficiently large, is irreducible (and non empty iff the class satisfies the condition which we define as 'admissibility'). In the unramified case, a similar result had been proven by Dunfield and Thurston using the classical invariant in the second homology group of G, H_2(G, \ZZ). We achieve our result showing that the stable classes are in bijection with the set of admissible classes $\e$.