- Algebraic surface
In

mathematics , an**algebraic surface**is analgebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface is therefore of complex dimension two (as acomplex manifold , when it isnon-singular ) and so of dimension four as asmooth manifold .The theory of algebraic surfaces is much more complicated than that of

algebraic curve s (including thecompact Riemann surface s, which are genuinesurface s of (real) dimension two). Many results were obtained, however, in theItalian school of algebraic geometry , and are up to 100 years old.Examples of algebraic surfaces include (κ is the

Kodaira dimension ):* κ= −∞: the projective plane,

quadric s in "P"^{3},cubic surface s,Veronese surface ,del Pezzo surface s,ruled surface s

* κ= 0 :K3 surface s,abelian surface s,Enriques surface s,hyperelliptic surface s

* κ= 1:Elliptic surface s

* κ= 2: 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 afunction field isomorphic to that of theprojective plane , being therational function s in two indeterminates. The cartesian product of two curves also provides examples.The

birational geometry of algebraic surfaces is rich, because ofblowing up (also known as amonoidal transformation ); under which a point is replaced by the "curve" of all limiting tangent directions coming into it (aprojective line ). Certain curves may also be blown "down", but there is a restriction (self-intersection number must be −1).Basic results on algebraic surfaces include the

Hodge index theorem , and the division into five groups of birational equivalence classes called theclassification of algebraic surfaces . The "general type" class, ofKodaira dimension 2, is very large (degree 5 or larger for a non-singular surface in "P"^{3}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 abirational invariant , becauseblowing up can add whole curves, with classes in "H"^{1,1}. It is known thatHodge cycle s are algebraic, and thatalgebraic equivalence coincides withhomological equivalence , so that "h"^{1,1}is an upper bound for ρ, the rank of theNéron-Severi group . Thearithmetic genus "p"_{"a"}is the difference:geometric genus − irregularity.

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

The

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

*http://www.freigeist.cc/gallery.html A gallery of algebraic surfaces*]

* [*http://www.singsurf.org/singsurf/SingSurf.html SingSurf*] an interactive 3D viewer for algebraic surfaces.

* [*http://www.mathematik.uni-kl.de/%7Ehunt/drawings.html Some beautiful algebraic surfaces*]

* [*http://www1-c703.uibk.ac.at/mathematik/project/bildergalerie/gallery.html Some more, with their respective ecuations*]

* [*http://www.bru.hlphys.jku.at/surf/index.html New Page on Algebraic Surfaces started in June 2008*]

*Wikimedia Foundation.
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