Normal polytopes and ellispoids

Abstract: Lattice polytopes are the combinatorial backbone of toric varieties. Many important properties of these varieties admit purely combinatorial description in terms of the underlying polytopes. These include normality and projective normality. On the other hand, there are geometric properties of polytopes of integer programming/discrete optimization origin, which can be used to deduce the aforementioned combinatorial properties: existence of unimodular triangulations or unimodular covers. In this talk we present the following recent results: (1) unimodular simplices in a lattice 3-polytope cover a neighborhood of the boundary if and only if the polytope is very ample, (2) the convex hull of lattice points in every ellipsoid in R^3 has a unimodular cover, and (3) for every d at least 5, there are ellipsoids in R^d, such that the convex hulls of the lattice points in these ellipsoids are not even normal. Part (3) answers a question of Bruns, Michalek, and the speaker. Chaiperson - Siamak Yassemi

Mathematics

Audience: advanced learners

IIT Bombay Virtual Commutative Algebra Seminar

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