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# A Note on Toric Varieties Associated with Moduli Spaces

Published:2010-12-31
Printed: Sep 2011
• James J. Uren,
Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3
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## 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. A. Tyurin used this construction on a certain collection of trinion decomposed surfaces to produce a variety $DM_g$, the so-called \emph{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 $g \geq 3.$
 MSC Classifications: 14M25 - Toric varieties, Newton polyhedra [See also 52B20] 52B20 - Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx]