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 2009 (vol 52 pp. 366)
 Gévay, Gábor

A Class of Cellulated Spheres with NonPolytopal Symmetries
We construct, for all $d\geq 4$, a cellulation of $\mathbb S^{d1}$.
We prove that these cellulations cannot be polytopal with maximal
combinatorial symmetry. Such nonrealizability phenomenon was first
described in dimension 4 by Bokowski, Ewald and Kleinschmidt, and,
to the knowledge of the author, until now there have not been any
known examples in higher dimensions. As a starting point for the
construction, we introduce a new class of (Wythoffian) uniform
polytopes, which we call duplexes. In proving our main result,
we use some tools that we developed earlier while studying perfect
polytopes. In particular, we prove perfectness of the duplexes;
furthermore, we prove and make use of the perfectness of another
new class of polytopes which we obtain by a variant of the socalled
$E$construction introduced by Eppstein, Kuperberg and Ziegler.
Keywords:CW sphere, polytopality, automorphism group, symmetry group, uniform polytope Categories:52B11, 52B15, 52B70 

4. CMB 2006 (vol 49 pp. 161)
 Agapito, José

Weighted BrianchonGram Decomposition
We give in this note a weighted version of Brianchon and Gram's
decomposition for a simple polytope. We can derive from this
decomposition the weighted polar formula of Agapito and a weighted
version of Brion's theorem, in a manner similar to Haase, where the
unweighted case is worked out. This weighted version of Brianchon
and Gram' decomposition
is a direct consequence of the ordinary BrianchonGram formula.
Category:52B 

5. 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 
