Definition
(Type of a log map.) Suppose \((C, f, \mathbf p)\) is a log map over a log point \(S = \Spec(Q\to k)\). It is the quadruple \(\tau = (G, \mathbf g, \sigma, u)\) where
- \(G = V(G) \cup E(G) \cup L(G)\) is the dual intersection graph of a source curve where vertices \(V(G)\) are the generic points of components, edges \(E(G)\) are the nodes and legs \(L(G)\) are the marked points
- \(\mathbf g:V(G) \to \mathbb N\) is the genus of a vertex
- \(\sigma:V(G)\cup E(G) \cup L(G) \to \Sigma(X)\) is the “cone function” which assigns to a point \(x\) the cone \(\sigma(x) = \left(\overline{\mathcal M}_{X, f(x)}\right)^\vee_{\mathbb R}\) in the tropicalization of \(X\), it is the piece of \(X\) to which \(x\) maps in the tropicalization
- \(u = \{u_p, u_q\}\) is the contact order of the marks and the nodes.
Attached to this construction is the basic monoid \(Q(\tau)\), which depends only \(G\), \(\sigma\) and \(u\). We emphasize that this is discrete data associated to a log map.