Zero scheme of a line-bundle section

Assume the Axiom of Choice as inherited from the affine quotient and gluing constructions (The Axiom of Choice). Let $X$ be a scheme, let $L$ be an invertible ${\mathcal{O}}_X$-module (Invertible sheaves) with dual $L^{-1}=L^{\vee }=\mathcal{H}o{m}_{{\mathcal{O}}_X}(L,{\mathcal{O}}_X)$, and let $s\in \mathrm{\Gamma }(X,L)$ be a global section.

The ideal of the section. The duality pairing $L^{-1}{\otimes }_{{\mathcal{O}}_X}L\to {\mathcal{O}}_X$ and $s$ induce the contraction map $c_s:L^{-1}⟶{\mathcal{O}}_X,\phi ⟼\phi (s),$ defined on an open $U\subseteq X$ by $\phi \mapsto \phi (s{\mathrm{\mid }}_U)$. Its image ${\mathcal{I}}_s=\mathrm{Im}(c_s)\subseteq {\mathcal{O}}_X$ is a quasi-coherent sheaf of ideals: $c_s$ is a morphism of quasi-coherent ${\mathcal{O}}_X$-modules and the image of such a morphism is quasi-coherent (Kernels and cokernels of quasi-coherent modules). The zero scheme of $s$ is the closed subscheme $Z(s)=V({\mathcal{I}}_s)\hookrightarrow X$ cut out by this ideal sheaf, in the sense of the affine quotient description of closed subschemes (Closed immersions are affine quotients and survive base change): on an affine open $U=\mathrm{Spec}A\subseteq X$ the intersection $Z(s)\cap U$ is $\mathrm{Spec}(A\mathrm{/}{I}_s)$ with $I_s=\mathrm{\Gamma }(U,{\mathcal{I}}_s)$.

Local equations. Let $U=\mathrm{Spec}A$ be an affine open on which $L$ is trivialised, $\phi :L{\mathrm{\mid }}_U\stackrel{\sim }{\to }{\mathcal{O}}_U$, and let $f=\phi (s{\mathrm{\mid }}_U)\in A$ be the corresponding local equation of $s$. Then under $\phi$ the map $c_s{\mathrm{\mid }}_U$ becomes multiplication by $f$, ${\mathcal{O}}_U\stackrel{\cdot f}{\to }{\mathcal{O}}_U,$ by the identification $L^{-1}{\mathrm{\mid }}_U\cong {\mathcal{O}}_U$ dual to $\phi$, so ${\mathcal{I}}_s{\mathrm{\mid }}_U=(f)$ and $Z(s)\cap U=\mathrm{Spec}(A\mathrm{/}(f))\mathrm{.}$ A different trivialisation ${\phi }^{\mathrm{\prime }}=\phi u$ with $u\in A^{\times }$ replaces $f$ by $uf$, hence does not change the ideal $(f)$; the local descriptions therefore glue along overlaps by the canonicity of the ideal $(f)$ (Gluing affine schemes along compatible open isomorphisms), and $Z(s)$ is well defined independently of trivialisations.

Nowhere-vanishing sections. If $s$ vanishes nowhere, then each local equation $f$ is a unit, so ${\mathcal{I}}_s={\mathcal{O}}_X$, the induced closed immersion $Z(s)\to X$ is an isomorphism onto the empty subscheme, and one says $Z(s)=\mathrm{\varnothing }$; this is the empty divisor case below.

Effective Cartier divisors. $Z(s)$ is an effective Cartier divisor — that is, ${\mathcal{I}}_s$ is an invertible sheaf of ideals, locally generated by one nonzerodivisor — precisely when for one (equivalently every) cover of $X$ by affine opens $U=\mathrm{Spec}A$ trivialising $L$, each local equation $f\in A$ of $s$ is a nonzerodivisor or a unit of $A$. If every local equation is a unit, $Z(s)=\mathrm{\varnothing }$ is the empty divisor; otherwise $Z(s)$ is a closed subscheme locally cut out by a nonzerodivisor wherever a non-unit equation occurs, and its ideal sheaf is invertible there. No hypothesis on $X$ beyond the trivialisations of $L$ enters.

• Consistency with the affine case. If $L={\mathcal{O}}_X$ and $s=f$ is a global function, then $c_s$ is multiplication by $f$ and $Z(s)=V(f)$ is the closed subscheme of zeros of $f$; this is the classical zero locus.
• Suppliers. The identifications $L^{-1}\otimes L\cong {\mathcal{O}}_X$ and the quasi-coherence of images are cited from Dual of a line bundle is its tensor inverse and Kernels and cokernels of quasi-coherent modules.