Related papers: On the birational geometry of matroids
We explore birational geometry of matroids by investigating automorphisms of their coarse Bergman fans. Combinatorial Cremona maps provide such automorphisms of Bergman fans which are not induced by matroid automorphisms. We investigate the…
We study the fan structure of Dressians $\Dr(d,n)$ and local Dressians $\Dr(\cM)$ for a given matroid $\cM$. In particular we show that the fan structure on $\Dr(\cM)$ given by the three term Pl\"ucker relations coincides with the structure…
We introduce the Bergman fan of a polymatroid and prove that the Chow ring of the Bergman fan is isomorphic to the Chow ring of the polymatroid. Using the Bergman fan, we establish the K\"ahler package for the Chow ring of the polymatroid,…
The tropical variety defined by linear equations with constant coefficients is the Bergman fan of the corresponding matroid. Building on a self-contained introduction to matroid polytopes, we present a geometric construction of the Bergman…
We study the Bergman complex B(M) of a matroid M: a polyhedral complex which arises in algebraic geometry, but which we describe purely combinatorially. We prove that a natural subdivision of the Bergman complex of M is a geometric…
We introduce the notion of real phase structure on rational polyhedral fans in Euclidean space. Such a structure consists of an assignment of affine spaces over $\mathbb{Z}/2\mathbb{Z}$ to each top dimensional face of the fan subject to two…
We prove that the proper amalgam of matroids $M_1$ and $M_2$ along their common restriction $N$ exists if and only if the tropical fibre product of Bergman fans ${B(M_1) \times_{B(N)} B(M_2)}$ is positive. We introduce tropical…
The theory of matroids or combinatorial geometries originated in linear algebra and graph theory, and has deep connections with many other areas, including field theory, matching theory, submodular optimization, Lie combinatorics, and total…
We define Chern-Schwartz-MacPherson (CSM) cycles of an arbitrary matroid. These are balanced weighted fans supported on the skeleta of the corresponding Bergman fan. In the case that the matroid arises from a complex hyperplane arrangement…
We study the positive Bergman complex B+(M) of an oriented matroid M, which is a certain subcomplex of the Bergman complex B(M) of the underlying unoriented matroid. The positive Bergman complex is defined so that given a linear ideal I…
This paper explores the birational geometry of a general Horrocks-Mumford quintic threefold, describing the set of all minimal models up to marked isomorphism, the movable fan (the way in which the nef cones of all these models are arranged…
Given a matroid $M$ one can define its Orlik-Solomon algebra $OS(M)$ and the Bergman fan $\Sigma_0(M)$. On the other hand to any rational polyhedral fan $\Sigma$ one can associate its tropical homology and cohomology groups…
This paper introduces Dirichlet matroids, a generalization of graphic matroids arising from electrical networks. We present four main theorems. First, we exhibit a matroid quotient involving geometric duals of networks embedded in surfaces…
Given a matroid M on the ground set E, the Bergman fan B(M), or space of M-ultrametrics, is a polyhedral complex in R^E which arises in several different areas, such as tropical algebraic geometry, dynamical systems, and phylogenetics.…
The Bergman fan of a matroid is the intersection of tropical hyperplanes defined by the circuits. A tropical basis is a subset of the circuits set that defines the Bergman fan. Yu and Yuster posed a question whether every simple regular…
We study moduli spaces of rational graphically stable tropical curves and a refinement given by radial alignment. Given a complete multipartite graph $\Gamma$, the moduli space of radially aligned $\Gamma$-stable tropical curves can be…
We classify closed curves isomorphic to the affine line in the complement of a smooth rational projective plane conic Q. Over a field of characteristic zero, we show that up to the action of the subgroup of the Cremona group of the plane…
Scattering diagrams arose in the context of mirror symmetry, Donaldson-Thomas theory, and integrable systems. We show that a consistent scattering diagram with minimal support cuts the ambient space into a complete fan. A special class of…
We show that if a group automorphism of a Cremona group of arbitrary rank is also a homeomorphism with respect to either the Zariski or the Euclidean topology, then it is inner up to a field automorphism of the base-field. Moreover, we show…
We study the generic fibre of the Hadamard product of linear spaces via matroid theory and tropical geometry. To do so, we introduce the flip product, a numerical invariant associated to a pair of matroids defined via the stable…