Multimatroids and Rational Curves with Cyclic Action

Abstract

We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids that naturally arise in topological graph theory. The perspective of moduli of curves provides a tropical framework for studying multimatroids, generalizing the previous connection between type-$A$ permutohedral varieties (Losev–Manin moduli spaces) and matroids, and the connection between type-$B$ permutohedral varieties and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of ${\mathbb {R}}$-multimatroids, a generalization of multimatroids, and we introduce the independence polytopal complex of a multimatroid, whose volume is identified with an intersection number on the moduli space. As an application, we give a combinatorial formula for a natural class of intersection numbers on the moduli space by relating to the volumes of independence polytopal complexes of multimatroids.