Definition
A stable logarithmic map \((C/S, \mathbf p, f)\) is a commutative diagram
where
- \(\pi:C\to S\) is a proper, logarithmically smooth and integral morphism of log schemes together with a tuple of sections \(\mathbf p = (p_1,...,p_k)\) of \(\underline \pi\) such that every geometric fiber of \(\pi\) is a reduced and connected curve, and if \(U\subset \underline C\) is the non-critical locus of \(\underline \pi\) then \(\overline{\mathcal M}_C|_U \simeq \underline \pi^*\overline{\mathcal M}_S\oplus \bigoplus^k_{i=1}p_{i,*}\mathbb N_S\)
- For every geometric point \(\overline s \to \underline S\), the restriction of \(\underline f\) to \(\underline C_{\overline s}\) together with \(\mathbf p\) is an ordinary stable map.