Definition
A family of tropical curves \(\Gamma \to \omega\) over a cone \(\omega = Q^\vee_{\mathbb R}\) in the Gross-Siebert sense is a triple \((G, \mathbf g, \ell)\) where
- \(G\) is a genus decorated curve with vertices, edges and legs
- \(\mathbf g:V(G)\to \mathbb N\) is the genus decoration
- \(\ell:Q^\vee \to \mathbb N\) is the length map, so \(\ell(E)\) is the length of the edge \(E \in E(G)\).
This is made into something actually resembling a family by building a cone complex \(\Gamma\) with one cone \(\omega_x\) per \(x \in V(G)\cup E(G) \cup L(G)\) in the following way:
- for a vertex \(v\), \(\omega_v = \omega\) (one copy of \(\omega\) per vertex)
- for an edge \(E\), \(\omega_E = \{(s, \lambda) ~\mid~ 0\leq \lambda \leq \ell(E)(s)\} \subseteq \omega \times \mathbb R_{\geq 0}\)
- for a leg \(L\in L(G)\), \(\omega_L = \omega \times \mathbb R_{\geq 0}\).
Then vertex cones \(\omega_v\) map to edge and leg cones \(\omega_E\) and \(\omega_L\) by embedding in the \(\lambda = 0\) component. It’s instructive to examine the case where \(\omega = \pt\), then the resulting cone complex is exactly the metric graph \(G\).