Global Sections Of The Ghost Sheaf, Torsors, And Line Bundles

lemma

See [Ogu18, page 274].

Lemma

Let \(\alpha:\mathcal M_X\to \mathcal O_X\) be an integral log structure on \(X\). Then each global section \(s\in \Gamma(X, \overline{\mathcal M}_X)\) corresponds to a \(\mathcal O_X^*\)-torsor and thus a line bundle \(\mathcal L_s\) on \(X\).

Proof

Since the log structure is integral, we have an exact sequence

\begin{align*} 1\to \mathcal O_X^*\xrightarrow{\lambda} \mathcal M^{\text{gp}}_X\xrightarrow{\pi}\overline{\mathcal M}_X^{\text{gp}} \to 0 \end{align*}

where \(\lambda\) is \(\alpha^{-1}|_{\alpha^{-1}(\mathcal O_X^*)}\) composed with the injection \(\mathcal M_X\hookrightarrow\mathcal M_X^{\text{gp}}\). Then for any global section \(s\in \Gamma(X,\overline{\mathcal M}_X)\), the sheaf \(\mathcal L_s^* = \pi^{-1}(s)\subset \mathcal M_X\) admits an \(\mathcal O^*_X\)-action which is simply transitive and at each \(x\in X\). Furthermore, the stalk \((\mathcal L_s^*)_x\) at least contains a copy of \(\mathcal O_X^*\) and hence is nonempty, so \(\mathcal L_s^*\) is an \(\mathcal O_X^*\)-torsor. Up to isomorphism this corresponds to a class in \(H^1(X, \mathcal O_X^*)\) and hence a line bundle \(\mathcal L_s\) on \(X\).

In fact, the map

\begin{align*} \Gamma(X, \overline{\mathcal M}_X^{\text{gp}}) \to \Pic(X);~ s\mapsto [\mathcal L_s] \end{align*}

is simply the boundary map of the long exact sequence on cohomology associated to the above short exact sequence.

Remark

The point here is that both an \(\mathcal O_X^*\)-torsor and a line bundle are defined up to isomorphism by patching data in \(\mathcal O_X^*\). This means that, while the \(\mathcal O_X^*\)-torsor \(\mathcal L^*_s = \pi^{-1}(s)\) isn’t literally a line bundle on \(X\), its isomorphism class uniquely determines \(\mathcal O_X^*\) gluing data, which in turn can be used to build a line bundle unique up to isomorphism.

Example

Let \(\mathcal M_X\) be the divisorial log structure on \(X\) corresponding to a smooth divisor \(D\). Then the inverse image of \(n\in \Gamma(X, \overline{\mathcal M}_X)\cong \mathbb N\) under the map \(\mathcal M_X\to \overline{\mathcal M_X}\) is the sheaf \(\mathcal L_n^*\) defined

\begin{align*} U&\mapsto \{f\in \mathcal M_X(U) \mid f \text{ vanishes to at least \(n\)th order on }D\}. \end{align*}

Let \(\mathcal O_X^*(-nD)\) denote the subsheaf of \(\mathcal O_X(-nD)\) consisting of “sections which are locally units away from \(D\)”, which makes sense once we identify \(\mathcal O_X(-nD)(V)\) with \(\mathcal O_X(V)\) for a sufficiently small neighborhood \(V\). This is precisely \(\mathcal L^*_n\) above, and the corresponding line bundle is \(\mathcal O_X(-nD)\).

Punchline: The line bundle associated to \(n\in \Gamma(X, \overline{\mathcal M}_X)\) is \(\mathcal O_X(-nD)\).

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