A Note on Toric Varieties Associated to Moduli Spaces

TL;DR

This paper reviews the construction of toric varieties from genus g Riemann surfaces with pants decompositions, discusses their properties, and clarifies that certain moduli space models are singular for genus three and above.

Contribution

It provides a concise review of toric varieties related to moduli spaces and clarifies the singularity properties of Tyurin's Delzant models for higher genus surfaces.

Findings

Tyurin's Delzant models are singular for genus g ≥ 3.

The paper details the moment polytopes of these toric varieties.

It confirms the construction of toric varieties from Riemann surfaces with pants decompositions.

Abstract

In this note we give a brief review of the construction of a toric variety $\mathcal{V}$ coming from a genus $g \geq 2$ Riemann surface $\Sigma^g$ equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey in \cite{JH1}. In \cite{T1} A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $DM_g$ -- the so-called Delzant model of moduli space -- for each genus $g.$ We conclude this note with some basic facts about the moment polytopes of the varieties $\mathcal{V}.$ In particular, we show that the varieties $DM_g$ constructed by Tyurin, and claimed to be smooth, are in fact singular for $g \geq 3.$