Proposition
Suppose \((C/S, \mathbf p, f)\) is a stable log curve with log target \((X/B, \mathcal M_X)\) where \(S\) is a log point. Let \(\underline C_0 \subset \underline C\) be an irreducible component of \(C\) corresponding to vertex \(v\in V(G)\) of the dual graph of \(C\). Let \(p_1,...,p_n\) be the marked points on \(C_0\) and \(q_1,...,q_m\) be the nodes contained in \(C_0\) with corresponding contact orders \(u_{p_i}, u_{q_j}\). Note that the contact order at the node is “oriented pointing away from the vertex \(v\)”. Then for any \(s\in \Gamma(X, \overline{\mathcal M}_X)\) giving rise to line bundle \(\mathcal L_s\),
\begin{align*} \deg \big((f^*\mathcal L_s)|_{C_0}\big) = -\sum_{i=1}^n \langle u_{p_i}, s\rangle - \sum_{j=1}^m \langle u_{q_j}, s\rangle. \end{align*}