Loci of curves with subcanonical points in low genus

TL;DR

This paper computes the classes of specific loci of genus 3 and 4 curves with special marked points in the moduli space, advancing understanding of subcanonical points and their geometric properties.

Contribution

It provides explicit calculations of the classes of loci with subcanonical points in low genus moduli spaces, linking to minimal strata of Abelian differentials.

Findings

Computed classes of hyperelliptic and non-hyperelliptic loci in genus 3

Determined class of genus 4 locus with even theta characteristic

Connected loci to minimal strata of Abelian differentials

Abstract

Inside the moduli space of curves of genus three with one marked point, we consider the locus of hyperelliptic curves with a marked Weierstrass point, and the locus of non-hyperelliptic curves with a marked hyperflex. These loci have codimension two. We compute the classes of their closures in the moduli space of stable curves of genus three with one marked point. Similarly, we compute the class of the closure of the locus of curves of genus four with an even theta characteristic vanishing with order three at a certain point. These loci naturally arise in the study of minimal strata of Abelian differentials.