A Note on Toric Varieties Associated to Moduli Spaces

Details A Note on Toric Varieties Associated to Moduli Spaces A Note on Toric Varieties Associated to Moduli Spaces

2008-11-28

arxiv.org/abs/0811.4728v1

Mathematics

Algebraic Geometry

7 pages, no figures, to appear in Canad. Math. Bull

Scientific paper

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 $g \geq 3.$

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