Combinatorial challenges in toric varieties

April 27 to May 1, 2009

at the

American Institute of Mathematics, San Jose, 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 2P 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:

Few smooth d-polytopes with N lattice points
by  Tristram Bogart, Christian Haase, Milena Hering, Benjamin Lorenz, Benjamin Nill, Andreas Paffenholz, Francisco Santos, and Hal Schenck
Convex normality of rational polytopes with long edges
by  Joseph Gubeladze
A simple combinatorial criterion for projective toric manifolds with dual defect
by  Alicia Dickenstein and Benjamin Nill