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 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 $\mathbb{R}$-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.

Figures (10)

• Figure 1: The fan $\Sigma_{B_2}$, with the cone corresponding to $\mathscr{C}= (\{{2}\} \subsetneq \{\bar{1},{2}\})$ shaded.
• Figure 2: The fan $\Sigma^\pi$ for $(E,\pi)$ as in \ref{['ex:nonuniform']}, with the cone corresponding to $\mathscr{C}= (\{{a}\} \subsetneq \{{a},{2}\})$ shaded.
• Figure 3: On the left, the normal complex $\mathrm{C}_{\Sigma^\pi, \ast}(D)$ from \ref{['ex:cubical']}. In the middle, the polytope $P_{\sigma_{\mathscr{C}},\ast}(D)$ and its bounding hyperplanes, where $\sigma_\mathscr{C}$ is the maximal cone associated with the chain $\mathscr{C}=\left(\{{1}\} \subseteq \{{1},{2}\}\right)$. On the right, $\mathrm{C}_{\Sigma^\pi, \ast}(D)$ is subdivided into simplices, each of volume 1.
• Figure 4: On the left, the normal complex $\mathrm{C}_{\Sigma^\pi, \ast}(D)$ from \ref{['ex:sumHnotcubical']}. On the right, the polytope $P_{\sigma_{\mathscr{C}},\ast}(D)$ and its bounding hyperplanes, where $\sigma_\mathscr{C}$ is the maximal cone associated with the chain $\mathscr{C}=\left(\{a\} \subseteq \{1,a\}\right)$. Note that the intersection of the bounding hyperplanes lies on the boundary of $\sigma_{\mathscr{C}}$.
• Figure 5: The independence polytopal complex of the multimatroid $\mathbf{M}$ of \ref{['ex:M12bar12']}

Theorems & Definitions (81)

• Theorem A
• Theorem B
• Remark 1.1
• Definition 2.2
• Remark 2.3
• Definition 2.5
• Example 2.6
• Example 2.7
• Remark 2.8
• Proposition 2.10