A Note on Toric Varieties Associated to Moduli Spaces
Abstract
In this note we give a brief review of the construction of a toric variety V coming from a genus g ≥ 2 Riemann surface g equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey in JH1. In T1 A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety DMg -- 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 V. In particular, we show that the varieties DMg constructed by Tyurin, and claimed to be smooth, are in fact singular for g ≥ 3.
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