Locus of curves with D_n-symmetry inside M_g

TL;DR

This paper classifies the irreducible components of the moduli space of genus g curves with dihedral symmetry by analyzing subgroup actions within the mapping class group, advancing understanding of symmetric curve loci.

Contribution

It provides a detailed classification of the loci of curves with D_n-symmetry inside the moduli space M_g, based on subgroup analysis in the mapping class group.

Findings

Identified irreducible components of M_g with D_n-symmetry.

Classified pairs of subgroups with specific fixed point properties.

Enhanced understanding of symmetric curves in moduli space.

Abstract

The aim of this paper is to determine the irreducible components of M_g(D_n), the locus inside M_g of the curves admitting an effective action by the dihedral group D_n. This is done by classifying pairs (H,H') of distinct subgroups of the mapping class group Map_g, such that both H and H' are isomorphic to D_n and the fixed point locus of H inside the Teichm\"uller space T_g is contained in the fixed point locus of H'.