Moduli space of meromorphic differentials with marked horizontal separatrices

TL;DR

This paper investigates the topology of moduli spaces of meromorphic differentials with marked horizontal separatrices, revealing their connected components and topological invariants, especially highlighting differences in hyperelliptic cases.

Contribution

It provides a classification of the connected components of these moduli spaces and introduces a topological invariant to distinguish them, extending understanding of flat geometry near the Deligne-Mumford boundary.

Findings

Number of connected components is at most two for genus > 0.

A simple topological invariant distinguishes the components.

Hyperelliptic cases exhibit unique component structures.

Abstract

We study framed translation surfaces corresponding to meromorphic differentials on compact Riemann surfaces, for which a horizontal separatrix is marked for each pole or zero. Such geometric structures naturally appear when studying flat geometry surfaces "near" the Deligne-Mumford boundary.We compute the number of connected components of the corresponding strata, and give a simple topological invariant that distinguishes them. In particular we see that for $g>0$, there are at most two such components, except in the hyperelliptic case.