On the birational geometry of matroids

TL;DR

This paper explores the birational geometry of matroids through Bergman fan isomorphisms, revealing that fine structure isomorphisms come from matroid isomorphisms, while Cremona automorphisms extend this understanding.

Contribution

It introduces Cremona automorphisms of Bergman fans and characterizes automorphism groups for specific classes of matroids, advancing the understanding of matroid birational geometry.

Findings

Isomorphisms respecting fine fan structure are induced by matroid isomorphisms.

Cremona automorphisms are introduced for coarse Bergman fan structures.

Automorphism group of the coarse structure is generated by matroid automorphisms and Cremona maps for certain matroids.

Abstract

This paper investigates isomorphisms of Bergman fans of matroids respecting different fan structures, which we regard as matroid analogs of birational maps. We show that isomorphisms respecting the fine fan structure are induced by matroid isomorphisms. We introduce Cremona automorphisms of the coarse structure of Bergman fans, which are not induced by matroid automorphisms. We show that the automorphism group of the coarse fan structure is generated by matroid automorphisms and Cremona maps in the case of rank 3 matroids which are not parallel connections and for modularly complemented matroids.