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

This workshop, sponsored by AIM and the NSF, 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.

The workshop will differ from typical conferences in some regards. Participants will be invited to suggest open problems and questions before the workshop begins, and these will be posted on the workshop website. These include specific problems on which there is hope of making some progress during the workshop, as well as more ambitious problems which may influence the future activity of the field. Lectures at the workshop will be focused on familiarizing the participants with the background material leading up to specific problems, and the schedule will include discussion and parallel working sessions.

The deadline to apply for support to participate in this workshop has passed.

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