Classification Of Log Structures Of Log Smooth Curves

Theorem

Let \(\pi:C\to S\) be a log smooth curve. Then the underlying map of schemes is a flat family of nodal curves with a collection of sections \(\{p_i\}\). The log structure \(\mathcal M_C\) behaves in three ways relative to the base \(S\).

1. At a generic point \(\eta \) of \(C\),

\begin{equation*} \mathcal M_{C,\eta} \cong \pi^*\mathcal M_{S, \pi(\eta)} \end{equation*}

1. At a node \(q\in C\), the ghost sheaf \(\overline{\mathcal M}_{C,q}\cong \mathcal M_{C,q}/\mathcal O_{X,q}^\times\) is generated by two sections \(x,y\in \mathcal M_{C,q}\) corresponding to local coordinates for the two branches meeting at \(q\) (the images \(\alpha_C(x),\alpha_C(y)\) are literally the functions which give \(\mathcal O_{C,q} \cong k[x,y]_{(x,y)}/(xy)\)) which satisfy the relation

\begin{equation*} xy = \pi^*(t) \end{equation*}

for some \(t\in \mathcal M_{S, \pi(q)}\).

1. At a marked section \(p_i\) the log structure is generated by a single monomial defining the divisor \(p_i(S)\subset C\).