Definition
(Type of a family of tropical maps.) The type of a family of tropical maps is the same data as the type of a stable log map, but read off from the tropical data.
Geometric ~
\(,\quad\) Tropical ~
Start with a family of tropical curves \(\Gamma \to \omega\) over a cone \(\omega\) together with a tropical map \(h:\Gamma \to \Sigma(X)\) from \(\Gamma\) to a cone complex \(\Sigma(X)\). We think of \(\Gamma\) here as a genus-decorated metric graph \((G, \mathbf g, \ell)\) rather than the cone complex associated to it. The type is then
\begin{align*} \tau = (G, \mathbf g, \sigma, \mathbf u) \end{align*}where
- \(G\) is the underlying graph
- \(\mathbf g:V(G)\to \mathbb N\) is the genus decorating function
- \(\sigma:V(G)\cup E(G) \cup L(G) \to \Sigma(X)\) is the map assigning each piece of graph data to the cone in which its image lands
- \(\mathbf u\) is the contact orders where \(u_E \in N_{\sigma(E)}\) is defined by \(h(v') - h(v) = \ell(E) \cdot u_E\) along each edge and \(u_L\in N_{\sigma(L)}\) is the outgoing slope of each leg.
Notice that the length \(\ell\) appearing as part of the data of \(\Gamma \)is forgotten.
Compared the the type of a stable log map, now \(\sigma(x)\) is the minimal cone containing the image of \(h(\omega_x)\) and the contact orders are the images of tangent vectors \((0,1)\) under \(h\) (which makes them much more interpretable in my opinion).