Definition
\((X, \mathcal M_X)\) a fine saturated log scheme with a decomposition of monoids \(\mathcal M_X = \mathcal M \oplus_{\mathcal O_X^\times} \mathcal P\). A puncturing of \(X\) is a submonoid \(\mathcal M^\circ_X \subset \mathcal M\oplus_{\mathcal O_X^\times} \mathcal P^{gp}\) such that \(\mathcal M_X \subset \mathcal M_X^\circ\) and
- The inclusion \(\mathcal M_X \hookrightarrow \mathcal M^\circ_X\) is a morphism of log structures on \(X\)
- Let \(x\in X\) be a geometric point and \(s_x\in \mathcal M^\circ_x\) some section which is not in \(\mathcal M_x\). Then for any \((m_x, p_x)\in \mathcal M_x\oplus \mathcal P_x^{gp}\), \(\alpha_{\mathcal M^\circ_X}(s_x) = \alpha_{\mathcal M}(m_x) = 0\).