Picard groups of moduli spaces of curves with symmetry

TL;DR

This paper investigates the structure of Picard groups of moduli spaces of smooth curves with automorphisms, showing finite generation under certain conditions and computing them explicitly in special cases, especially for hyperelliptic curves.

Contribution

It establishes finite generation of Picard groups for moduli spaces with automorphisms and computes these groups explicitly in specific cases, advancing understanding of their algebraic structure.

Findings

Picard groups are finitely generated under mild conditions

Explicit computations of Picard groups for certain moduli spaces

Finite abelian level covers of hyperelliptic loci have finitely generated Picard groups

Abstract

We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the first part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the second part of the paper, we show that finite abelian level covers of the hyperelliptic locus in the moduli space of smooth curves have finitely generated Picard groups. We also compute the Picard groups of the moduli spaces of hyperelliptic curves of compact type.