We prove the hard Lefschetz theorem and the Hodge-Riemann relations for a commutative
ring associated to an arbitrary matroid M. We use the Hodge-Riemann relations to resolve a …

We introduce certain torus-equivariant classes on permutohedral varieties which we call
“tautological classes of matroids” as a new geometric framework for studying matroids …

In a recent paper, the first author proved the log-concavity of the coefficients of the
characteristic polynomial of a matroid realizable over a field of characteristic 0, answering a …

We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to
algebraic intersection points with expected multiplicities, provided that the tropicalizations …

This article is an introduction to matroid theory aimed at algebraic geometers. Matroids are
combinatorial abstractions of linear subspaces and hyperplane arrangements. Not all …

We show that the weights on a tropical variety can be recovered from the tropical scheme
structure proposed in [GG16], so there is a well-defined Hilbert–Chow morphism from a …

We show how to equip the cone complexes of toroidal embeddings with additional structure
that allows to define a balancing condition for weighted subcomplexes. We then proceed to …

An enumerative problem on a variety V is usually solved by reduction to intersection theory
in the cohomology of a compactification of V. However, if the problem is invariant under a …

Let X be a smooth and projective threefold with a simple normal crossings divisor D. We
construct the Donaldson–Thomas theory of the pair enumerating ideal sheaves on X relative …

We propose a geometric interpretation of Block and Göttsche's refined tropical curve
counting invariants in terms of virtual χ− y specializations of motivic measures of …