Type Of Family Of Tropical Maps And Decorated Type

Definition

1. The type of a family of tropical maps \(h:\Gamma\to \Sigma(X)\) over \(Q^\vee_{\mathbb R} \in \mathbf{Cones}\) is the quadruple \(\tau = (G, \mathbf{g}, \boldsymbol{\sigma}, \mathbf{u})\) consisting of
   • the associated genus decorated graph \((G, \mathbf{g})\) (i.e. \(G\) is a graph with vertices \(V(G)\), edges \(E(G)\) and legs \(L(G)\) and \(g:V(G)\to \mathbb N\) assigns a genus to each vertex, the genus of the corresponding irreducible component when \(G\) is the dual graph of a curve)
   • the map \(\boldsymbol \sigma:V(G)\cup E(G) \cup L(G)\to \Sigma(X)\) maps each generic point, node or marked point \(x\) to the minimal cone \(\tau \in \Sigma(X)\) containing \(\omega_x\), the cone in \(\Gamma\) associated to \(x\) (note that if \(E\) is a leg or edge incident to a vertex \(v\), then there is an inclusion of faces \(\boldsymbol \sigma(v) \subset \boldsymbol \sigma(E)\) in the presentation of \(\Sigma(X)\))
   • the contact orders \(\mathbf u = \{u_p, u_q\}\)
2. For a type \(\tau\) of a family of tropical maps, \(\Aut(\tau)\) denotes the subset of automorphisms of \(G\) commuting with the maps \(\mathbf g\), \(\boldsymbol \sigma\), \(\mathbf u\).
3. Given a type \(\tau\) of a family of tropical maps, the associated basic monoid \(Q(\tau)\) is the dual of the monoid \(Q^\vee\) defined below, depending only on \(G\), \(\boldsymbol \sigma\) and \(\mathbf u\).
4. If in addition we have a map \(A:V(G)\to H^+_2(X)\) assigning a curve class to each irreducible component, then we call \(\tilde \tau = (\tau, A)\) the decorated type of a family of tropical maps, the pair \((h, A)\) a decorated family of tropical maps and \(|A| = \sum_{v\in V(G)} A(v)\) the total curve class of \(A\).

The monoid \(Q^\vee\) is defined \[Q^\vee = \left\{\big((V_\eta)_\eta, (e_q)_q\big) \in \bigoplus_\eta P^\vee \oplus \bigoplus_q \mathbb N ~ \middle | ~ \forall q: V_{\eta_2} - V_{\eta_1} = e_qu_q\right\}.\] This is apparently the “moduli cone of tropical curves of fixed combinatorial type”. It should be the dual of the basic monoid.

We say that \(\tau\) is rigid if \(Q(\tau) = \mathbb N\). (Definition 3.6).