Definition
A pre-stable marked log curve \((C/S, p)\) is a proper log-smooth integral morphism \(\pi:C\to S\) of fine saturated log schemes of relative dimension \(1\) (necessarily a flat family) together with a tuple of sections \(p = (p_1,...,p_n)\) of \(\underline \pi\) such that every fiber of \(\underline \pi\) is a reduced and connected curve, and if \(U\subseteq \underline C\) is the non critical locus of \(\underline \pi\) then \[\overline{\mathcal M}_{C}|_U \cong \pi^*\overline{\mathcal M}_S \oplus \bigoplus_{i} p_i^*\mathbb N_S.\]