Editing Algebraic surface (section)

==  Classification by the Kodaira dimension ==
{{Main|Enriques–Kodaira classification}}
In the case of dimension one, varieties are classified by only the [[topological genus]], but, in dimension two, one needs to distinguish the [[arithmetic genus]] <math>p_a</math> and the [[geometric genus]] <math>p_g</math> because one cannot distinguish birationally only the topological genus. Then, [[Irregularity of a surface|irregularity]] is introduced for the classification of varieties. A summary of the results (in detail, for each kind of surface refers to each redirection), follows:

Examples of algebraic surfaces include (κ is the [[Kodaira dimension]]):

* κ&thinsp;=&thinsp;−∞: the [[complex projective plane|projective plane]], [[quadric]]s in '''P'''<sup>3</sup>, [[cubic surface]]s, [[Veronese surface]], [[del Pezzo surface]]s, [[ruled surface]]s
* κ&thinsp;=&thinsp;0 : [[K3 surface]]s, [[abelian surface]]s, [[Enriques surface]]s, [[hyperelliptic surface]]s
* κ&thinsp;=&thinsp;1: [[elliptic surface]]s
* κ&thinsp;=&thinsp;2: [[surface of general type|surfaces of general type]].

For more examples see the [[list of algebraic surfaces]].

The first five examples are in fact [[birationally equivalent]]. That is, for example, a cubic surface has a [[function field of an algebraic variety|function field]] isomorphic to that of the [[projective plane]], being the [[rational function]]s in two indeterminates. The Cartesian product of two curves also provides examples.