The diagonal of tropical matroid varieties and cycle intersections

TL;DR

This paper develops an intersection product for tropical cycles on matroid varieties, enabling intersection theory on moduli spaces of tropical curves and extending to smooth varieties, with applications to cycle pull-backs and rational equivalence.

Contribution

It introduces a well-behaved intersection product on matroid varieties and extends it to smooth tropical varieties, facilitating intersection theory on tropical moduli spaces.

Findings

Defined an intersection product via diagonal cutting

Enabled intersection computations on tropical moduli spaces

Extended intersection theory to smooth tropical varieties

Abstract

We define an intersection product of tropical cycles on matroid varieties (via cutting out the diagonal) and show that it is well-behaved. In particular, this enables us to intersect cycles on moduli spaces of tropical rational marked curves $M_{0,n}$ and $M_{0,n}(\R^r, d)$. This intersection product can be extended to smooth varieties (whose local models are matroid varieties). We also study pull-backs of cycles and rational equivalence.