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 $g \geq 3.$

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 $X$.