Classification of Surfaces

The classification of surfaces is one of the many great classical results of algebraic geometry. In this section we collect some notes necessary to support the classification.

Definition:

Let $D$ be a divisor and $\iota_D$ the (possibly rational) map given by $D$. Define $\kappa(D) = \dim(\iota_D)$, so that for a variety $X$ the Kodaira dimension $\kappa(X) = \max_{n\in\mathbb N}\kappa(nK_X)$ for $K_X$ the cannonical divisor of $X$. We follow the convention that $\dim(\emptyset) = -\infty$.

We show the

Lemma:

Let $D$ be an effective divisor. Then $h^0(\mathcal O_X(D))\geq 1$.

Lemma:

An effective divisor in a connected variety $X$ has $\kappa(D)\geq 1$.

Proof. Suppose $D$ is an effective divisor which induces a birational map $\iota:X\to \mathbb P^n$. Restricting to the open $U$ on which $\iota$ is defined gives that $D$ is the vanishing locus of a (sufficiently general) section $s$ of $i^*\mathcal O_{\mathbb P^n}(1)$.