The canonical divisor of a smooth projective surface

Assume the Axiom of Choice, inherited from the dualizing-bundle supplier and the Euler-characteristic and Serre-duality suppliers below (The Axiom of Choice). Let $k$ be a field and let $X$ be an integral (Integral schemes), smooth (Smoothness over a field by geometric regularity), projective (Projective morphisms before Proj) $k$-scheme of pure dimension two (Chain dimension and the empty-space convention). Such an $X$ is an integral regular projective surface over $k$ (Intersection numbers of Cartier divisors on a smooth projective surface): it is of finite type over $k$ and smoothness makes every local ring regular, so all the intersection-theoretic and cohomological constructions of that item apply.

Let ${\omega }_X={\bigwedge }^2{\mathrm{\Omega }}_{X\mathrm{/}k}^1$ be the dualizing line bundle (Dualizing line bundle and trace datum of a smooth projective variety), an invertible ${\mathcal{O}}_X$-module, and recall that all its cohomology groups are finite-dimensional and the Euler characteristic $\chi (X,-)$ of Euler characteristic of a coherent sheaf is defined on it.

Canonical divisor. A canonical divisor on $X$ is a Cartier divisor $K_X$ (Cartier divisor) whose associated invertible sheaf satisfies ${\mathcal{O}}_X(K_X)\cong {\omega }_X$ (Invertible sheaf of cartier divisor).

One exists. The generic stalk of ${\omega }_X$ is one-dimensional over the function field $K(X)$, so ${\omega }_X$ has a rational section $s$ (Rational section line bundle); by Rational sections of line bundles are Cartier divisors the divisor $K_X={\mathrm{div}}_C(s)$ is a well-defined Cartier divisor and the canonical section $1_{K_X}$ identifies ${\mathcal{O}}_X(K_X)\cong {\omega }_X$ carrying $1_{K_X}$ to $s$. Conversely, for every Cartier divisor $D$ the canonical section $1_D$ is a rational section of ${\mathcal{O}}_X(D)$ with ${\mathrm{div}}_C(1_D)=D$.

Any two canonical divisors are linearly equivalent (Linear equivalence cartier divisors), and every Cartier divisor linearly equivalent to a canonical divisor is canonical. Indeed, if $D,D^{\mathrm{\prime }}$ are Cartier divisors with ${\mathcal{O}}_X(D)\cong {\mathcal{O}}_X(D^{\mathrm{\prime }})\cong {\omega }_X$, transport the canonical rational section $1_D$ across an isomorphism $\phi :{\mathcal{O}}_X(D)\to {\mathcal{O}}_X(D^{\mathrm{\prime }})$. There is a unique $g\in K(X{)}^{\times }$ with $\phi (1_D)=g{1}_{D^{\mathrm{\prime }}}$, since two rational sections of one invertible sheaf differ by a meromorphic unit (Rational section line bundle), and comparing local equations in trivializations shows ${\mathrm{div}}_C(g{1}_{D^{\mathrm{\prime }}})={\mathrm{div}}_C(g)+D^{\mathrm{\prime }}$ (Cartier divisor). Divisors of sections are preserved by a sheaf isomorphism, and ${\mathrm{div}}_C(1_D)=D$ and ${\mathrm{div}}_C(1_{D^{\mathrm{\prime }}})=D^{\mathrm{\prime }}$ by Rational sections of line bundles are Cartier divisors, so $D-D^{\mathrm{\prime }}={\mathrm{div}}_C(g)$ is principal and $D\sim D^{\mathrm{\prime }}$; the converse is immediate from the same computation.

Basic properties.

1. The intersection numbers $K_X\cdot D$ and $D\cdot K_X$ depend only on the linear equivalence classes of $K_X$ and $D$ and are therefore independent of the chosen canonical divisor (Intersection numbers of Cartier divisors on a smooth projective surface, The surface intersection product is symmetric and bilinear); in particular $K_X\cdot D=D\cdot K_X$. For an invertible ${\mathcal{O}}_X$-module $\mathcal{L}$ one writes $\mathcal{L}\cdot K_X:=\mathcal{L}\cdot {\omega }_X$ and $K_X\cdot \mathcal{L}:={\omega }_X\cdot \mathcal{L}$; these depend only on the classes.
2. Serre duality (Serre duality for locally free sheaves on a smooth projective variety) gives $\chi (X,{\omega }_X)=\chi (X,{\mathcal{O}}_X)$ and, for every invertible ${\mathcal{O}}_X$-module $\mathcal{L}$, the identity $\chi (X,\mathcal{L}\otimes {\omega }_X)=\chi (X,{\mathcal{L}}^{\vee })$: dualizing $E={\mathcal{O}}_X$ and $E={\mathcal{L}}^{\vee }$ pairs $H^q$ with $H^{2-q}$ and preserves the alternating sum because the pairings are perfect and all groups are finite-dimensional. Equivalently $\chi (X,{\mathcal{O}}_X(K_X+D))=\chi (X,{\mathcal{O}}_X(-D))$ for every Cartier divisor $D$ on $X$.
3. The Euler characteristic of the structure sheaf is the integer $\chi (X,{\mathcal{O}}_X)=h^0(X,{\mathcal{O}}_X)-h^1(X,{\mathcal{O}}_X)+h^2(X,{\mathcal{O}}_X)$ with $h^q={\dim }_k{H}^q(X,{\mathcal{O}}_X)$; by property 2 it equals $\chi (X,{\omega }_X)$. The surface arithmetic genus is $p_a(X)=\chi (X,{\mathcal{O}}_X)-1$ (Vakil, §18.4.4). No topological description of this integer is used on this page. The expression becomes $1-h^1+h^2$ when $H^0(X,{\mathcal{O}}_X)=k$; integrality alone over an arbitrary field does not imply that condition.