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Search: MSC category 52B20 ( Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] )

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1. CMB 2010 (vol 54 pp. 561)

Uren, James J.
A Note on Toric Varieties Associated with Moduli Spaces
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 $g \geq 3.$

Categories:14M25, 52B20

2. CMB 2005 (vol 48 pp. 414)

Kaveh, Kiumars
Vector Fields and the Cohomology Ring of Toric Varieties
Let $X$ be a smooth complex projective variety with a holomorphic vector field with isolated zero set $Z$. From the results of Carrell and Lieberman there exists a filtration $F_0 \subset F_1 \subset \cdots$ of $A(Z)$, the ring of $\c$-valued functions on $Z$, such that $\Gr A(Z) \cong H^*(X, \c)$ as graded algebras. In this note, for a smooth projective toric variety and a vector field generated by the action of a $1$-parameter subgroup of the torus, we work out this filtration. Our main result is an explicit connection between this filtration and the polytope algebra of $X$.

Keywords:Toric variety, torus action, cohomology ring, simple polytope,, polytope algebra
Categories:14M25, 52B20

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