Length Of Edge In Tropical Family Or Of A Node

Construction

Let \(C/S\) be a log smooth curve over a log point \(S = \Spec(Q\to k)\) with \(Q\) a sharp, fine, saturated monoid. Log smoothness forces \(\overline{\mathcal M}_{C, q} = Q\oplus_{\mathbb N} \mathbb N^2\) where the map \(\mathbb N\to Q\) is given by \(1\mapsto \rho_q\) for some nonzero element \(\rho_q\) of \(Q\); etale locally the node is \(xy = f\) for \(\overline f = \rho_q\) in \(\overline{\mathcal M}_S = Q\) . The length of \(E_q\), the edge asscociated to \(q\) in the tropicalization of \(C\), is

\begin{align*} \ell(E_q) = \langle \rho_q, - \rangle:Q^\vee \to \mathbb N. \end{align*}

In this view the element \(\rho_q\) is the length of \(E_q\), after identifying \(Q = Q^{\vee \vee}\). To realize it as a length, first move over to \(\mathbb R_{\geq 0}\) in three steps (abbreviating \(\ell(E_q)\) as \(\ell\)):

1. Group extension: since \(Q\) is a sharp, fine saturated monoid, \(Q^{gp}\) is a lattice and \(Q\) embeds in it as a full dimensional submonoid. Same holds for \(Q^\vee\). Thus additivity forces a unique extension of \(\ell\) to the lattice \(N = (Q^\vee)^{gp}\), \(\ell^{gp}:N\to \mathbb Z\) by defining \(\ell^{gp}(a - b) = \ell(a) - \ell(b)\).
2. Now tensor with \(\mathbb R\) to get \(\ell^{gp}_{\mathbb R}:N_{\mathbb R}\to \mathbb R\).
3. Now intersect with \(Q^{\vee}\). Since \(\ell^{gp}(Q^\vee) = \ell(Q^\vee) \subseteq \mathbb N\), \(\ell^{gp}_{\mathbb R}(Q^{\vee} \cap N_{\mathbb R}) \subseteq \mathbb R_{\geq 0}\). This gives us a map \(\ell_{\mathbb R}:Q^\vee_{\mathbb R} \to \mathbb R_{\geq 0}\).

This monoid \(Q^\vee_{\mathbb R}\) is the cone \(\omega\) over which the tropical family \(h:\Gamma\to \omega\) lives. Take any \(s\in \omega\) and the fiber \(h^{-1}(s)\) is a copy of the dual graph \(G\) of \(C\). In it there is an edge \(E_q\), and the length of that edge is exactly

\begin{align*} \ell(E_q)(s) = \langle \rho_q, s\rangle \in \mathbb R_{\geq 0}. \end{align*}

This is precisely the length function which makes \(G\) into a metric graph.