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We establish a connection between the algebraic geometry of the type permutohedral toric variety and the combinatorics of delta‐matroids. Using this connection, we compute the volume and lattice point counts of type generalized permutohedra. Applying tropical Hodge theory to a new framework of “tautological classes of delta‐matroids,” modeled after certain vector bundles associated to realizable delta‐matroids, we establish the log‐concavity of a Tutte‐like invariant for a broad family of delta‐matroids that includes all realizable delta‐matroids. Our results include new log‐concavity statements for all (ordinary) matroids as special cases.

Polymatroids are combinatorial abstractions of subspace arrangements in the same way that matroids are combinatorial abstractions of hyperplane arrangements. By introducing augmented Chow rings of polymatroids, modeled after augmented wonderful varieties of subspace arrangements, we generalize several algebro-geometric techniques developed in recent years to study matroids. We show that intersection numbers in the augmented Chow ring of a polymatroid are determined by a matching property known as the Hall–Rado condition, which is new even in the case of matroids.

We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.