Classes of Weierstrass points on genus 2 curves

TL;DR

This paper investigates the classes of loci of genus 2 curves with multiple Weierstrass points, providing recursive formulas and generating functions to describe their closures within moduli spaces, advancing understanding of hyperelliptic classes.

Contribution

It introduces a recursive method and generating functions to describe the classes of Weierstrass point loci in genus 2 moduli spaces, extending to all n within stable and compact type curves.

Findings

Recursive description of locus classes in moduli space

Generating functions for boundary strata classes

Extension to all n in compact type moduli space

Abstract

We study the codimension n locus of curves of genus 2 with n distinct marked Weierstrass points inside the moduli space of genus 2, n-pointed curves, for n <= 6. We give a recursive description of the classes of the closure of these loci inside the moduli space of stable curves. For n<= 4, we express these classes using a generating function over stable graphs indexing the boundary strata of moduli spaces of pointed stable curves. Similarly, we express the closure of these classes inside the moduli space of curves of compact type for all n. This is a first step in the study of the structure of hyperelliptic classes in all genera.