Cell decompositions of moduli space, lattice points and Hurwitz problems

TL;DR

This paper explores cell decompositions of the moduli space of Riemann surfaces, linking them to Hurwitz problems, and provides methods to compute lattice points and volumes that reveal deep geometric insights.

Contribution

It introduces a new cell decomposition approach with linear structures, enabling effective calculation of lattice points and volumes related to the moduli space.

Findings

Cell decompositions form rational convex polytopes with natural integer points.

Methods to compute lattice points and volumes across all cells.

Calculations encode deep geometric information about the moduli space.

Abstract

In this article we describe cell decompositions of the moduli space of Riemann surfaces and their relationship to a Hurwitz problem. The cells possess natural linear structures and with respect to this they can be described as rational convex polytopes which come equipped with natural integer points and a volume form. We show how to effectively calculate the number of lattice points and the volumes over all the cells and that these calculations contain deep information about the moduli space.