Combinatorial Cremona automorphisms and Coxeter arrangement matroids

TL;DR

This paper studies the automorphisms of matroid Bergman fans, introducing combinatorial Cremona maps that extend beyond matroid automorphisms, and characterizes their structure especially for Coxeter arrangement matroids.

Contribution

It introduces combinatorial Cremona maps as new automorphisms of Bergman fans and provides a realizability criterion for matroids with multiple Cremonas.

Findings

Automorphisms of Bergman fans include combinatorial Cremona maps not induced by matroid automorphisms.

A realizability criterion for matroids with two different Cremonas is established.

For Coxeter arrangement matroids, automorphisms are generated by matroid automorphisms and at most one Cremona map.

Abstract

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 structure of matroids allowing combinatorial Cremona maps and prove a realizability criterion in the presence of two different Cremonas. We also prove that for all matroids associated to Coxeter arrangements the group of coarse automorphisms of the Bergman fan is generated by the matroid automorphisms and at most one combinatorial Cremona map.