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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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