Definition
Let \(X\) be a scheme and \(D\subset X\) be a closed subset of pure codimension 1. Set \(U = X\setminus D\) and denote by \(j:U \hookrightarrow X\) the inclusion. Then the divisorial log structure induced by \(D\) is the log structure \(\mathcal M_{(X,D)}\) defined
\begin{align*} \mathcal M_{(X,D)} := \left(j_*\mathcal O_U^\times\right)\cap \mathcal O_X \end{align*}and taking
\begin{align*} \alpha_X:\mathcal M_{(X,D)} \hookrightarrow \mathcal O_X \end{align*}to be the inclusion. Informally, \(\mathcal M_{(X,D)}\) consists of all functions which are invertible away from \(D\).