#
Combinatorial challenges in toric varieties

April 27 to May 1, 2009
at the

American Institute of Mathematics,
Palo Alto, California

organized by

Joseph Gubeladze,
Christian Haase,
and Diane Maclagan

## Original Announcement

This workshop will be devoted to a
selection of problems on lattice polytopes that arise in the theory of
toric varieties. Besides structural results, we will work on search
strategies and computational approaches to these questions.
The first of these is the question of which lattice polytopes *P* have the
property that every lattice point in the dilation 2*P* is a sum of two
lattice points in *P*. With 2 replaced by *k* for all *k>2*, this asks whether
the corresponding polarized toric variety is projectively normal. An
important open case is whether this property holds whenever the toric
variety is smooth.

A closely related question on a smooth projectively normal toric variety
is whether it is necessarily defined by quadrics. A stronger property
would be the existence of quadratic regular unimodular triangulation of
the underlying polytope, in which case the homogenous coordinate ring of
the variety is even Koszul. We will also search for broader combinatorial
interpretations of this property.

If the polytope *P* is sufficiently dilated such a (quadratic/regular)
unimodular triangulation is known to exist, but the proof is not
effective. We will seek a concrete bound on the constant *c* for which the
polytope *cP* has these properties. It is unclear whether there is a
uniform bound depending only on the dimension.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop: