Kodaira dimension and the Enriques classification of surfaces

Kodaira dimension measures how much the pluricanonical system sees: nothing, a point, a curve, or the whole surface for $\kappa=-\infty,0,1,2$.Suggest a slogan

The Kodaira dimension of a smooth projective variety $X$ is \[ \kappa(X) = \operatorname{trdeg}_k \bigoplus_{n \geq 0} H^0(X, \OO_X(nK_X)) - 1 , \] set to $-\infty$ when every plurigenus vanishes. It is a birational invariant, so for surfaces it is computed on a minimal model.

(Zariski desingularization of surfaces)

Every projective surface $Y$ admits a desingularization by a finite sequence that alternates normalizations and blowups of maximal ideals. The resulting smooth surface is projective.

(Minimal resolution of normal surface singularities)

Let $Y$ be a normal projective surface. There exists a unique minimal resolution $\varphi \colon X \to Y$ such that $X$ is smooth projective, $\varphi$ is an isomorphism over $\reg Y$, and every other desingularization $\varphi' \colon X' \to Y$ with the same two properties factors as $\varphi' = \varphi \circ \sigma$ where $\sigma \colon X' \to X$ is a composition of blowups of points over the exceptional locus of $\varphi$.

(Enriques classification)

A minimal smooth projective surface $X$ over $\CC$ falls into exactly one of:

$\kappa(X) = -\infty$ iff $|12K_X| = \varnothing$ iff $X$ is $\PP^2$ or a minimal ruled surface;
$\kappa(X) = 0$ iff $|12K_X| = \{0\}$ iff $X$ is K3, Enriques, abelian, or bielliptic;
$\kappa(X) = 1$ iff $X$ is a properly elliptic surface;
$\kappa(X) = 2$ iff $X$ is of general type.

In particular $\kappa = -\infty$ are the rational and ruled surfaces, $\kappa = 0$ are the four classes above, $\kappa = 1$ are elliptic, and $\kappa = 2$ are general type.

The organising idea is that $\kappa$ measures how much of the surface the pluricanonical systems can see: nothing at $-\infty$, a point at $0$, a curve at $1$, the whole surface at $2$. For curves the same invariant gives the trichotomy $g=0$, $g=1$, $g \geq 2$, and the surface classification is its two-dimensional analogue.

The $\kappa = 0$ row is the one to be able to separate, by $(p_g, q)$: K3 has $(1,0)$ with $K \sim 0$ and simply connected; Enriques has $(0,0)$ with $2K \sim 0$, and is a free quotient of a K3 by an involution; abelian has $(1,2)$; bielliptic has $(0,1)$ and is a quotient of a product of elliptic curves.

Minimality is what makes the statement clean, and blowing up is exactly the operation that leaves $\kappa$ alone while changing $K^2$ — which is why the classification is stated for minimal models and read off as a birational statement.

Over an algebraically closed field of characteristic $p > 0$ the classification of Bombieri and Mumford has the same four values of $\kappa$, but the $\kappa = 0$ and $\kappa = 1$ classes change: in characteristic $2$ and $3$ there are quasi-hyperelliptic surfaces and quasi-elliptic fibrations, and in characteristic $2$ there are non-classical Enriques surfaces with $h^1(\OO_X) = 1$. So the list above is a theorem for complex surfaces, equivalently for surfaces over an algebraically closed field of characteristic $0$.

Discuss the classification of surfaces.

What is the Kodaira dimension?

Which surfaces have Kodaira dimension zero?

Dependencies

T-SRFCASTCastelnuovo's contractibility criterionuses
D-SRFRULEDRational and ruled surfaces, and the Hirzebruch surfacesuses

Backlinks

FE-SRFFERMATQUARTICThe Fermat quartic is a K3 surface containing $48$ linesrelated-to

What links to this

Blowups and the classification of surfaces