Editing Algebraic surface (section)

== Properties ==

The [[ample line bundle#Intersection theorem|'''Nakai criterion''']] says that:
:A Divisor ''D'' on a surface ''S'' is ample if and only if ''D<sup>2</sup> > 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 <math>\mathcal{D}(S)</math> be the abelian group consisting of all the divisors on ''S''. Then due to the [[intersection number|intersection theorem]]
:<math>\mathcal{D}(S)\times\mathcal{D}(S)\rightarrow\mathbb{Z}:(X,Y)\mapsto X\cdot Y</math>
is viewed as a [[quadratic form]]. Let
:<math>\mathcal{D}_0(S):=\{D\in\mathcal{D}(S)|D\cdot X=0,\text{for all } X\in\mathcal{D}(S)\}</math>
then <math>\mathcal{D}/\mathcal{D}_0(S):=Num(S)</math> becomes to be a '''numerical equivalent class group''' of ''S'' and
:<math>Num(S)\times Num(S)\mapsto\mathbb{Z}=(\bar{D},\bar{E})\mapsto D\cdot E</math>
also becomes to be a quadratic form on <math>Num(S)</math>, where <math>\bar{D}</math> is the image of a divisor ''D'' on ''S''. (In the below the image <math>\bar{D}</math> is abbreviated with ''D''.)

For an ample line bundle ''H'' on ''S'', the definition
:<math>\{H\}^\perp:=\{D\in Num(S)|D\cdot H=0\}.</math>
is used in the surface version of the '''Hodge index theorem''':
:for <math>D\in\{\{H\}^\perp|D\ne0\}, D\cdot D < 0</math>, i.e. the restriction of the intersection form to <math>\{H\}^\perp</math> 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 conjectures|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 '''P'''<sup>3</sup> lies in it, for example).

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

:geometric genus &minus; irregularity.

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