Mathematics Colloquia and Seminars

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Description

Over three decades ago, Sturmfels established a direct connection between the existence of quadratic Gröbner bases and regular flag unimodular triangulations. We solve a famous question posed by Herzog and Hibi about quadratic Gröbner bases for matroid toric ideals: we show that the base polytopes of the Fano plane and its dual do not have regular flag unimodular triangulations. Our proof relies on several novel tools: a lemma connecting the 1-skeleton of a lattice polytope to the lattice points in its dilations, an encoding with Boolean formulas and SAT solvers, and symmetry-breaking arguments. These tools can also decide whether a given lattice polytope has a unimodular or regular unimodular flag triangulation. Using them, we classify all full-dimensional (0,1)-polytopes of dimension at most five according to the existence of unimodular, flag unimodular, and regular flag unimodular (quadratic) triangulations. Among the consequences: there is an infinite family of polytopes with a flag unimodular triangulation whose toric ring is not Koszul, and the proportion of normal (0,1)-polytopes tends to zero as the dimension grows.