71.7 Effective Cartier divisors and invertible sheaves
Since an effective Cartier divisor has an invertible ideal sheaf (Definition 71.6.1) the following definition makes sense.
Definition 71.7.1. Let S be a scheme. Let X be an algebraic space over S and let D \subset X be an effective Cartier divisor with ideal sheaf \mathcal{I}_ D.
The invertible sheaf \mathcal{O}_ X(D) associated to D is defined by
\mathcal{O}_ X(D) = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{I}_ D, \mathcal{O}_ X) = \mathcal{I}_ D^{\otimes -1}.
The canonical section, usually denoted 1 or 1_ D, is the global section of \mathcal{O}_ X(D) corresponding to the inclusion mapping \mathcal{I}_ D \to \mathcal{O}_ X.
We write \mathcal{O}_ X(-D) = \mathcal{O}_ X(D)^{\otimes -1} = \mathcal{I}_ D.
Given a second effective Cartier divisor D' \subset X we define \mathcal{O}_ X(D - D') = \mathcal{O}_ X(D) \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(-D').
Some comments. We will see below that the assignment D \mapsto \mathcal{O}_ X(D) turns addition of effective Cartier divisors (Definition 71.6.6) into addition in the Picard group of X (Lemma 71.7.3). However, the expression D - D' in the definition above does not have any geometric meaning. More precisely, we can think of the set of effective Cartier divisors on X as a commutative monoid \text{EffCart}(X) whose zero element is the empty effective Cartier divisor. Then the assignment (D, D') \mapsto \mathcal{O}_ X(D - D') defines a group homomorphism
\text{EffCart}(X)^{gp} \longrightarrow \mathop{\mathrm{Pic}}\nolimits (X)
where the left hand side is the group completion of \text{EffCart}(X). In other words, when we write \mathcal{O}_ X(D - D') we may think of D - D' as an element of \text{EffCart}(X)^{gp}.
Lemma 71.7.2. Let S be a scheme. Let X be an algebraic space over S. Let D \subset X be an effective Cartier divisor. Then for the conormal sheaf we have \mathcal{C}_{D/X} = \mathcal{I}_ D|D = \mathcal{O}_ X(D)^{\otimes -1}|_ D.
Proof.
Omitted.
\square
Lemma 71.7.3. Let S be a scheme. Let X be an algebraic space over S. Let D_1, D_2 be effective Cartier divisors on X. Let D = D_1 + D_2. Then there is a unique isomorphism
\mathcal{O}_ X(D_1) \otimes _{\mathcal{O}_ X} \mathcal{O}_ X(D_2) \longrightarrow \mathcal{O}_ X(D)
which maps 1_{D_1} \otimes 1_{D_2} to 1_ D.
Proof.
Omitted.
\square
Definition 71.7.4. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{L} be an invertible sheaf on X. A global section s \in \Gamma (X, \mathcal{L}) is called a regular section if the map \mathcal{O}_ X \to \mathcal{L}, f \mapsto fs is injective.
Lemma 71.7.5. Let S be a scheme. Let X be an algebraic space over S. Let f \in \Gamma (X, \mathcal{O}_ X). The following are equivalent:
f is a regular section, and
for any x \in X the image f \in \mathcal{O}_{X, \overline{x}} is not a zerodivisor.
for any affine U = \mathop{\mathrm{Spec}}(A) étale over X the restriction f|_ U is a nonzerodivisor of A, and
there exists a scheme U and a surjective étale morphism U \to X such that f|_ U is a regular section of \mathcal{O}_ U.
Proof.
Omitted.
\square
Note that a global section s of an invertible \mathcal{O}_ X-module \mathcal{L} may be seen as an \mathcal{O}_ X-module map s : \mathcal{O}_ X \to \mathcal{L}. Its dual is therefore a map s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_ X. (See Modules on Sites, Lemma 18.32.4 for the dual invertible sheaf.)
Definition 71.7.6. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{L} be an invertible sheaf. Let s \in \Gamma (X, \mathcal{L}). The zero scheme of s is the closed subspace Z(s) \subset X defined by the quasi-coherent sheaf of ideals \mathcal{I} \subset \mathcal{O}_ X which is the image of the map s : \mathcal{L}^{\otimes -1} \to \mathcal{O}_ X.
Lemma 71.7.7. Let S be a scheme. Let X be an algebraic space over S. Let \mathcal{L} be an invertible \mathcal{O}_ X-module. Let s \in \Gamma (X, \mathcal{L}).
Consider closed immersions i : Z \to X such that i^*s \in \Gamma (Z, i^*\mathcal{L})) is zero ordered by inclusion. The zero scheme Z(s) is the maximal element of this ordered set.
For any morphism of algebraic spaces f : Y \to X over S we have f^*s = 0 in \Gamma (Y, f^*\mathcal{L}) if and only if f factors through Z(s).
The zero scheme Z(s) is a locally principal closed subspace of X.
The zero scheme Z(s) is an effective Cartier divisor on X if and only if s is a regular section of \mathcal{L}.
Proof.
Omitted.
\square
Lemma 71.7.8. Let S be a scheme. Let X be an algebraic space over S.
If D \subset X is an effective Cartier divisor, then the canonical section 1_ D of \mathcal{O}_ X(D) is regular.
Conversely, if s is a regular section of the invertible sheaf \mathcal{L}, then there exists a unique effective Cartier divisor D = Z(s) \subset X and a unique isomorphism \mathcal{O}_ X(D) \to \mathcal{L} which maps 1_ D to s.
The constructions D \mapsto (\mathcal{O}_ X(D), 1_ D) and (\mathcal{L}, s) \mapsto Z(s) give mutually inverse maps
\left\{ \begin{matrix} \text{effective Cartier divisors on }X
\end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{pairs }(\mathcal{L}, s)\text{ consisting of an invertible}
\\ \mathcal{O}_ X\text{-module and a regular global section}
\end{matrix} \right\}
Proof.
Omitted.
\square
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