A Note on Toric Varieties Associated to Moduli Spaces

2008/11/28 by James J. Uren, Uren, James J.
Mathematics · #14M25 #52B20 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #math.AG #msc:14M25 #msc:52B20

paper · pdf · doi:10.48550/arxiv.0811.4728

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

arxiv created 2008/11/28 · openalex publication_date 2008/11/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 Σ^g equipped with a trinion, or pair of pants, decomposition. This was outlined by J. Hurtubise and L. C. Jeffrey in \citeJH1. In \citeT1 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 V. In particular, we show that the varieties DM_g constructed by Tyurin, and claimed to be smooth, are in fact singular for g ≥ 3.

Related

• Moduli spaces of Delzant polytopes and symplectic toric manifolds
• Irreducible components of moduli spaces of maps to smooth projective toric varieties in genus 0
• Moduli space of weighted pointed stable curves and toric topology of Grassmann manifolds
• Toric degenerations of weight varieties and applications
• Toric Degenerations of GIT Quotients, Chow Quotients, and M_0,n