Given a family of stable log maps \((C/S, f, p)\) mapping to \(X/B\), we obtain a corresponding diagram of cone complexes
A fiber of \(\Sigma(\pi)\) is a graph and a restriction of \(\Sigma(f)\) to such a fiber can be viewed as mapping a tropical curve into \(\Sigma(X)\).
The reference for this is all of Section 2.5 of “Decomposition of Degenerate GW Invariants”. What follows is a discussion of Proposition 2.25 from that section with an example.
Let \(S = \Spec(Q\to k)\) be a log point with monoid \(Q\) and let \((C/S, p)\) be a family of marked log curves (no map to \(X\) yet).
Proposition
The tropicalization \[\Sigma(\pi):\Sigma(C)\to \Sigma(S) = Q^\vee_{\mathbb R}\] of \((C/S, p)\) is naturally a family of tropical curves \((G, g, \ell)\) over \(Q^\vee_{\mathbb R}\).
Proof
(Sketch) We take \(G\) to be the dual intersection graph of \(C\) which means
- \(V(G)\) is the set of generic points \(\eta\) of \(C\)
- \(E(G)\) is the set of nodes of \(C\)
- \(L(G)\) is the set of marks of \(C\).
- The genus \(g(v_\eta)\)of a vertex \(v_\eta\) is the geometric genus of \(C_\eta = \overline{\{\eta\}}\)
- The length \(\ell(E_q) \) of an edge \(E_q\) associated to a node \(q\) is the element \(\rho_q\in Q \), thought of as a map \(Q^\vee\to \mathbb N \) given by evaluation at \(\rho_q\). This element is the one coming from the structure of the ghost sheaf of \(C\)at \(q\).
Looking at the above proof, the only thing to compute before we can write down an honest tropical curve are the elements \(\rho_q \). These essentially determine the log structure of \(C\) (see Remark 1.2 in Log GW paper).