Séminaire Lotharingien de Combinatoire, 82B.45 (2019), 12 pp.

Christopher Eur

Divisors on matroids and their volumes

Abstract. 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 shifted rank-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.

Received: November 15, 2018. Accepted: February 17, 2019. Final version: April 1, 2019.

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