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Combinatorics Seminar: Divisors on matroids and their volumes September 17, 2018 (12:00 PM PDT - 01:00 PM PDT)
Parent Program: --
Location: UC Berkeley Math (Evans Hall 939)
Speaker(s) Christopher Eur (University of California, Berkeley)
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The classical volume polynomial in algebraic geometry measures the degrees of ample (and nef) divisors on a smooth projective variety. We introduce an analogous volume polynomial for matroids, give a complete combinatorial formula, and show that it is a valuation under matroid polytope subdivisions. For a realizable matroid, we thus obtain an explicit formula for the classical volume polynomial of the associated wonderful compactification; in particular, we obtain another formula for volumes of generalized permutohedra. We then introduce a new invariant called the volume of a matroid as a particular specialization of its volume polynomial, and discuss its algebro-geometric and combinatorial properties in connection to graded linear series on blow-ups of projective spaces.

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