Permutohedral complexes 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 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.

Wonderful compactifications and rational curves with cyclic action

“…First, analogously to the case of , we show in Proposition 5.2 that this fan can be identified with a moduli space of ‘tropical -curves’. And second, analogously to the way in which the permutohedron is the normal polytope of the fan of Losev–Manin space , we show in Proposition 5.4 that the polytopal complex constructed in [CDH+22] is a normal complex of , in the sense developed by Nathanson–Ross in [NR21]. This gives a more geometric interpretation of the correspondence between the boundary strata of and the faces of that was proven combinatorially in our previous work.…”

“…In this section, we review the definition and necessary properties of the moduli space introduced in [CDH+22], and we prove that it is a wonderful compactification of a product arrangement in . Throughout, we assume that .…”

“…The underlying curve C in such an object is an ‘ r -pinwheel curve’, which is a rational curve consisting of a central projective line from which r equal-length chains of projective lines (‘spokes’) emanate. This curve is equipped with an order- r automorphism , as well as marked points as follows: two distinct fixed points and of ; n labeled r -tuples of points satisfying for each i and j , where we allow and ; an additional labeled r -tuple satisfying for each , whose elements are distinct from one another as well as from and . These marked points are subject to a stability condition, the details of which can be found in [CDH+22, Section 2.1]. We refer to each tuple as a ‘light orbit’ of and the tuple as the ‘heavy orbit’.…”

“…In [CDH+22, Theorem 3.5], a fine moduli space for stable -curves is constructed, whose B -points correspond to families of stable -curves over the base scheme B as defined in [CDH+22, Definition 2.5]. More precisely, there is a connected component for any choice of primitive r th root of unity , all of which are isomorphic to one another, and the moduli space is the disjoint union of these connected components.…”

“…The moduli space of rational curves with cyclic action was constructed in our previous work [CDH+22] as a generalization of Losev and Manin’s moduli space of rational curves with weighted marked points. In particular, the Losev–Manin space , introduced in [LM00], is a toric variety whose associated polytope is the permutohedron , and the torus-invariant subvarieties of have a modular interpretation as ‘boundary strata’, so one obtains an inclusion- and dimension-preserving bijection between the boundary strata of and the faces of .…”