Adjunction formula for effective divisors on a smooth projective surface

Given: a field $k$, an integral smooth projective surface $X$ over $k$, a canonical divisor $K_X$ with ${\mathcal{O}}_X(K_X)\cong {\omega }_X$, and a nonzero effective Cartier divisor $C\subseteq X$.

$X$ is proper over $k$ and Noetherian; the intersection product on invertible sheaves and Cartier divisors is defined by the alternating sum of Intersection numbers of Cartier divisors on a smooth projective surface, is symmetric and $\mathbb{Z}$-bilinear, and depends only on the linear equivalence classes of its entries (The surface intersection product is symmetric and bilinear). For an effective Cartier divisor $C$ and any Cartier divisor $D$ one has $C\cdot D={\mathrm{deg}}_C({\mathcal{O}}_X(D){\mathrm{\mid }}_C)$ with ${\mathrm{deg}}_C(\mathcal{N})=\chi (C,\mathcal{N})-\chi (C,{\mathcal{O}}_C)$ (Intersection with a curve is the degree of the restriction, Degree of an invertible sheaf on a proper one-dimensional scheme).

Serre duality: for every invertible ${\mathcal{O}}_X$-module $E$ and every $q$ the cup-product pairing $H^q(X,E)\times H^{2-q}(X,E^{\vee }\otimes {\omega }_X)\to k$ is perfect and all groups are finite-dimensional (Serre duality for locally free sheaves on a smooth projective variety). Consequently $\chi (X,E)=\chi (X,E^{\vee }\otimes {\omega }_X)$ and $\chi (X,{\omega }_X)=\chi (X,{\mathcal{O}}_X)$; moreover ${\mathcal{O}}_X(K_X)\cong {\omega }_X$, so ${\mathcal{O}}_X(K_X+D)\cong {\omega }_X\otimes {\mathcal{O}}_X(D)$ and $({\omega }_X\otimes {\mathcal{O}}_X(D){)}^{\vee }\cong {\mathcal{O}}_X(-K_X-D)$ for every Cartier divisor $D$ (The canonical divisor of a smooth projective surface, Dual of a line bundle is its tensor inverse, Tensor product of sheaves of modules, Invertible sheaves).

Structure sequence and additivity: for the nonzero effective Cartier divisor $C$ there is a short exact sequence of coherent ${\mathcal{O}}_X$-modules $0\to {\mathcal{O}}_X(-C)\to {\mathcal{O}}_X\to i_{\ast }{\mathcal{O}}_C\to 0$ (Effective Cartier divisors give a short exact sequence); the Euler characteristic is additive on short exact sequences of coherent modules on the proper scheme $X$ (Euler characteristic is additive in short exact sequences); and $\chi (X,i_{\ast }{\mathcal{O}}_C)=\chi (C,{\mathcal{O}}_C)$ (Euler characteristic of a coherent sheaf, the projection-formula identification of Intersection with a curve is the degree of the restriction in the effective case). Hence $\chi (X,{\mathcal{O}}_X)-\chi (X,{\mathcal{O}}_X(-C))=\chi (C,{\mathcal{O}}_C)$.

The divisor $-K_X-C$ is a Cartier divisor and ${\mathcal{O}}_X(-K_X-C)\cong ({\omega }_X\otimes {\mathcal{O}}_X(C){)}^{\vee }$, while ${\mathcal{O}}_X(-C)\cong {\mathcal{O}}_X(C{)}^{\vee }$; the canonical divisor is a Cartier divisor and linear equivalence may be replaced by the associated invertible sheaves in all intersection computations (The canonical divisor of a smooth projective surface, Cartier divisor, Invertible sheaf of cartier divisor).

The Axiom of Choice is inherited from the Serre-duality and Euler-characteristic suppliers of [F2] and [F3]; the divisor $C$ and the canonical divisor are given data.