American Institute of Mathematics, Palo Alto, California
Joseph Gubeladze, Christian Haase, and Diane Maclagan
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 schedule.
A report on the workshop activities.