Algebraic surface

Birational geometry of surfaces

The birational geometry of algebraic surfaces is rich, because of blowing up (also known as a monoidal transformation), under which a point is replaced by the curve of all limiting tangent directions coming into it (a projective line). Certain curves may also be blown down, but there is a restriction (self-intersection number must be −1).

Castelnuovo's Theorem

One of the fundamental theorems for the birational geometry of surfaces is Castelnuovo's theorem. This states that any birational map between algebraic surfaces is given by a finite sequence of blowups and blowdowns.

Properties

The Nakai criterion says that:

A Divisor D on a surface S is ample if and only if D² > 0 and for all irreducible curve C on S D•C > 0.

Ample divisors have a nice property such as it is the pullback of some hyperplane bundle of projective space, whose properties are very well known. Let ${\displaystyle \mathcal{D}(S)}$ be the abelian group consisting of all the divisors on S. Then due to the intersection theorem

${\displaystyle \mathcal{D}(S)\times \mathcal{D}(S)\to \mathbb{Z}:(X,Y)\mapsto X\cdot Y}$

is viewed as a quadratic form. Let

${\displaystyle {\mathcal{D}}_0(S):=\{D\in \mathcal{D}(S)|D\cdot X=0,\text{for all }X\in \mathcal{D}(S)\}}$

then ${\displaystyle \mathcal{D}/{\mathcal{D}}_0(S):=Num(S)}$ becomes to be a numerical equivalent class group of S and

${\displaystyle Num(S)\times Num(S)\mapsto \mathbb{Z}=(\overline{D},\overline{E})\mapsto D\cdot E}$

also becomes to be a quadratic form on ${\displaystyle Num(S)}$, where ${\displaystyle \overline{D}}$ is the image of a divisor D on S. (In the below the image ${\displaystyle \overline{D}}$ is abbreviated with D.)

For an ample line bundle H on S, the definition

${\displaystyle \{H{\}}^{\perp }:=\{D\in Num(S)|D\cdot H=0\}.}$

is used in the surface version of the Hodge index theorem:

for ${\displaystyle D\in \{\{H{\}}^{\perp }|D\ne 0\},D\cdot D<0}$, i.e. the restriction of the intersection form to ${\displaystyle \{H{\}}^{\perp }}$ is a negative definite quadratic form.

This theorem is proven using the Nakai criterion and the Riemann-Roch theorem for surfaces. The Hodge index theorem is used in Deligne's proof of the Weil conjecture.

Basic results on algebraic surfaces include the Hodge index theorem, and the division into five groups of birational equivalence classes called the classification of algebraic surfaces. The general type class, of Kodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in P³ lies in it, for example).

There are essential three Hodge number invariants of a surface. Of those, h^1,0 was classically called the irregularity and denoted by q; and h^2,0 was called the geometric genus p_g. The third, h^1,1, is not a birational invariant, because blowing up can add whole curves, with classes in H^1,1. It is known that Hodge cycles are algebraic and that algebraic equivalence coincides with homological equivalence, so that h^1,1 is an upper bound for ρ, the rank of the Néron-Severi group. The arithmetic genus p_a is the difference

geometric genus − irregularity.

This explains why the irregularity got its name, as a kind of 'error term'.

Riemann-Roch theorem for surfaces

The Riemann-Roch theorem for surfaces was first formulated by Max Noether. The families of curves on surfaces can be classified, in a sense, and give rise to much of their interesting geometry.