1. CMB 2016 (vol 60 pp. 510)
 Haase, Christian; Hofmann, Jan

Convexnormal (Pairs of) Polytopes
In 2012 Gubeladze (Adv. Math. 2012)
introduced the notion of $k$convexnormal polytopes to show
that
integral polytopes all of whose edges are longer than $4d(d+1)$
have
the integer decomposition property.
In the first part of this paper we show that for lattice polytopes
there is no difference between $k$ and $(k+1)$convexnormality
(for
$k\geq 3 $) and improve the bound to $2d(d+1)$. In the second
part we
extend the definition to pairs of polytopes. Given two rational
polytopes $P$ and $Q$, where the normal fan of $P$ is a refinement
of
the normal fan of $Q$.
If every edge $e_P$ of $P$ is at least $d$ times as long as the
corresponding face (edge or vertex) $e_Q$ of $Q$, then $(P+Q)\cap
\mathbb{Z}^d
= (P\cap \mathbb{Z}^d ) + (Q \cap \mathbb{Z}^d)$.
Keywords:integer decomposition property, integrally closed, projectively normal, lattice polytopes Categories:52B20, 14M25, 90C10 

2. 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 socalled \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 

3. 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 
