Intersection numbers of Cartier divisors on a smooth projective surface

Assume the Axiom of Choice, inherited from the Euler-characteristic supplier below (The Axiom of Choice). Let $k$ be a field and let $X$ be an integral (Integral schemes), regular (embedding dimension and regular local ring), projective (Projective morphisms before Proj) $k$-scheme of pure dimension two (Chain dimension and the empty-space convention). A smooth projective surface over $k$ is an instance: for a finite-type $k$-scheme, smoothness over $k$ makes every local ring regular (Smoothness over a field by geometric regularity), and a smooth projective surface is integral by hypothesis here.

The scheme $X$ is proper over $k$ (Projective morphisms are proper), integral, and of finite type over the field $k$, hence locally Noetherian (Locally Noetherian and Noetherian schemes); the structure sheaf, its dual, all invertible sheaves (Invertible sheaves) and all their tensor products and duals are coherent ${\mathcal{O}}_X$-modules (Coherent module sheaves, Dual of a line bundle is its tensor inverse), so the Euler characteristic $\chi (X,-)$ of Euler characteristic of a coherent sheaf is defined on all sheaves appearing below and takes values in $\mathbb{Z}$.

The pairing on line bundles. For invertible ${\mathcal{O}}_X$-modules $\mathcal{L}$ and $\mathcal{M}$ define $\mathcal{L}\cdot \mathcal{M}:=\chi (X,{\mathcal{O}}_X)-\chi (X,{\mathcal{L}}^{\vee })-\chi (X,{\mathcal{M}}^{\vee })+\chi (X,{\mathcal{L}}^{\vee }{\otimes }_{{\mathcal{O}}_X}{\mathcal{M}}^{\vee })\in \mathbb{Z},$ the tensor product being that of Tensor product of sheaves of modules.

The pairing on Cartier divisors. For Cartier divisors $C$ and $D$ on $X$ (Cartier divisor) with associated invertible sheaves ${\mathcal{O}}_X(C)$ and ${\mathcal{O}}_X(D)$ (Invertible sheaf of cartier divisor) define $C\cdot D:={\mathcal{O}}_X(C)\cdot {\mathcal{O}}_X(D)\in \mathbb{Z}\mathrm{.}$

Basic properties.

1. Isomorphism invariance. The value depends only on the isomorphism classes of $\mathcal{L}$ and $\mathcal{M}$: duals and tensor products are functorial under isomorphism (Dual of a line bundle is its tensor inverse, Tensor product of sheaves of modules) and the Euler characteristic is an invariant of isomorphism classes (Euler characteristic of a coherent sheaf). Hence the pairing is a well-defined map $\mathrm{Pic}(X)\times \mathrm{Pic}(X)\to \mathbb{Z}$ with $\mathrm{Pic}(X)$ the Picard group of Picard group of a scheme.
2. Symmetry and the structure sheaf. The defining expression is symmetric in $\mathcal{L}$ and $\mathcal{M}$, so $\mathcal{L}\cdot \mathcal{M}=\mathcal{M}\cdot \mathcal{L}$. Substituting $\mathcal{M}={\mathcal{O}}_X$ gives $\mathcal{L}\cdot {\mathcal{O}}_X=\chi (X,{\mathcal{O}}_X)-\chi (X,{\mathcal{L}}^{\vee })-\chi (X,{\mathcal{O}}_X)+\chi (X,{\mathcal{L}}^{\vee })=0$, and equally ${\mathcal{O}}_X\cdot \mathcal{L}=0$.
3. Linear equivalence. If $C^{\mathrm{\prime }}$ is linearly equivalent to $C$ and $D^{\mathrm{\prime }}$ to $D$, then ${\mathcal{O}}_X(C^{\mathrm{\prime }})\cong {\mathcal{O}}_X(C)$ and ${\mathcal{O}}_X(D^{\mathrm{\prime }})\cong {\mathcal{O}}_X(D)$: linear equivalence of Cartier divisors is vanishing of the class in $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)$, and for the integral $X$ the rule $D\mapsto [{\mathcal{O}}_X(D)]$ induces an isomorphism $\mathrm{CaDiv}(X)\mathrm{/}{\mathrm{Prin}}_C(X)\stackrel{\sim }{\to }\mathrm{Pic}(X)$ (On an integral scheme, Cartier divisors modulo principal divisors compute the Picard group), with addition of divisors corresponding to tensor product (Addition of Cartier divisors is tensor product of their sheaves). By part 1 the value $C\cdot D$ depends only on these isomorphism classes, so $C^{\mathrm{\prime }}\cdot D^{\mathrm{\prime }}=C\cdot D$.

The pairing is studied in The surface intersection product is symmetric and bilinear, where it is proved to be a symmetric $\mathbb{Z}$-bilinear form on $\mathrm{Pic}(X)$; for an effective Cartier divisor $C\subseteq X$ the restriction-degree identity of Intersection with a curve is the degree of the restriction identifies the value with the degree of a restricted line bundle, and The intersection matrix of a point blowup of a regular surface computes it on a point blowup.