- 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]
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