A Note on Toric Varieties Associated with Moduli Spaces

DOI10.4153/CMB-2010-109-4zbMATH Open1258.14059arXiv0811.4728MaRDI QIDQ3095130

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Publication date: 28 October 2011

Published in: (Search for Journal in Brave)

Abstract: In this note we give a brief review of the construction of a toric variety $mathcalV$ coming from a genus $ggeq2$ Riemann surface $Sigm{a}^g$ 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 $D{M}_g$ -- 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 $mathcalV.$ In particular, we show that the varieties $D{M}_g$ constructed by Tyurin, and claimed to be smooth, are in fact singular for $ggeq3.$

Full work available at URL: https://arxiv.org/abs/0811.4728

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