Definition Of A Log Structure

Definition

Let \(X\) be a scheme and consider the sheaf of monoids \(\mathcal O_X\) with the monoid structure given by multiplication.

• A pre-log structure on \(X\) consists of a sheaf of monoids \(\mathcal M_X\) on \(X\) in addition to a homomorphism of sheaves of monoids

\begin{align*} \alpha_X:\mathcal M_X\to \mathcal O_X. \end{align*}

• Then \(\mathcal M_X\) is a log structure if in addition the restriction

  \begin{align*} \alpha_X|_{\alpha^{-1}(\mathcal O_X^\times)}:]\alpha^{-1}_X(\mathcal O^\times_X) \to \mathcal O_X^\times \end{align*}

is an isomorphism. A scheme \(X\) endowed with a log structure \((\mathcal M_X, \alpha_X)\) is called a log scheme.

• A morphism \(f:X\to Y\) of log schemes consists of a morphism of the underlying schemes \(f:X\to Y\) and a morphism of sheaves of monoids

\begin{align*} f^\flat:f^{-1}\mathcal M_Y\to \mathcal M_X \end{align*}

which fits into the commutative diagram

• The ghost sheaf \(\overline{\mathcal M}_X := \mathcal M_X/\mathcal O_X^\times\) is the cokernel of the \(\alpha^{-1}_X\) restricted to \(\mathcal O_X^\times\), yielding a short exact sequence

\begin{align*} 1 \to \mathcal O_X^\times \xrightarrow{\alpha^{-1}_X}\mathcal M_X \to \overline{\mathcal M}_X\to 0. \end{align*}

As indicated by the notation in the short exact sequence above, we write \(\overline{\mathcal M}_X\) additively.