A tropical intersection product in matroidal fans

TL;DR

This paper develops a new intersection product for tropical cycles within matroidal fans, linking matroid operations with tropical modifications, and applies it to tropical geometry realizability questions.

Contribution

It introduces a novel intersection product on tropical cycles in matroidal fans, generalizing previous work and providing an alternative intersection method not relying on Cartier divisors.

Findings

Constructed an intersection product on tropical cycles in matroidal fans.

Connected matroid deletion and restriction with tropical modifications.

Applied the product to questions of tropical geometry realizability.

Abstract

We construct an intersection product on tropical cycles contained in the Bergman fan of a matroid. To do this we first establish a connection between the operations of deletion and restriction in matroid theory and tropical modifications as defined by Mikhalkin. This product generalises the product of Allermann and Rau, and Allermann and also provides an alternative procedure for intersecting cycles which is not based on intersecting with Cartier divisors. Also, we simplify the definition in the case of one dimensional fan cycles in two dimensional matroidal fans and given an application of the intersection product to realisability questions in tropical geometry.