We study the connection between multimatroids and moduli spaces of rational curves with cyclic action. Multimatroids are generalizations of matroids and delta-matroids introduced by Bouchet, which naturally arise in topological graph theory. The vantage point 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 (Batyrev-Blume moduli spaces) and delta-matroids. Specifically, we equate a combinatorial nef cone of the moduli space with the space of R-multimatroids, a slight 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, for the generating set of the Chow ring of the moduli space consisting of all psi-classes and their pullbacks along certain forgetful maps, we give a combinatorial formula for their intersection numbers by relating to the volumes of independence polytopal complexes of multimatroids.

Hurwitz correspondences are certain multivalued self-maps of the moduli space M 0,N . They arise in the study of Thurston's topological characterization of rational functions. We consider the dynamics of Hurwitz correspondences and ask: On which compactifications of M 0,N should they be studied? We compare a Hurwitz correspondence H across various modular compactifications of M 0,N , and find a weighted stable curves compactification X † N that is optimal for its dynamics. We use X † N to show that the kth dynamical degree of H is the absolute value of the dominant eigenvalue of the pushforward induced by H on a natural quotient of H 2k (M 0,N ).

Let $\unicode[STIX]{x1D719}$ be a post-critically finite branched covering of a two-sphere. By work of Koch, the Thurston pullback map induced by $\unicode[STIX]{x1D719}$ on Teichmüller space descends to a multivalued self-map—a Hurwitz correspondence ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$—of the moduli space ${\mathcal{M}}_{0,\mathbf{P}}$. We study the dynamics of Hurwitz correspondences via numerical invariants called dynamical degrees. We show that the sequence of dynamical degrees of ${\mathcal{H}}_{\unicode[STIX]{x1D719}}$ is always non-increasing and that the behavior of this sequence is constrained by the behavior of $\unicode[STIX]{x1D719}$ at and near points of its post-critical set.

No abstract

No abstract