- List of algebraic surfaces
This is a list of named (classes of)

algebraic surface s andcomplex surface s. The notation κ stands for theKodaira dimension , which divides surfaces into four coarse classes.**Algebraic and complex surfaces***

abelian surface s (κ = 0) Two dimensional abelian varieties.

*algebraic surface s

* Barlow surfaces General type, simply connected.

* [*http://enriques.mathematik.uni-mainz.de/docs/Ebarthsextic.shtml Barth sextic*] A degree-6 surface in "P"^{3}with 65 nodes.

* [*http://enriques.mathematik.uni-mainz.de/docs/Ebarthdecic.shtml Barth decic*] A degree-10 surface in "P"^{3}with 345 nodes.

* Beauville surfaces General type

* bielliptic surfaces (κ = 0) Same as hyperelliptic surfaces.

* Bordiga surfaces A degree-6 embedding of the projective plane into "P"^{4}defined by the quartics through 10 points in general position.

* Burniat surfaces General type

* Campedelli surfaces General type

* Castelnuovo surfaces General type

* Catanese surfaces General type

* class VII surfaces κ = −∞, non-algebraic.

* Cayley surface Rational. A cubic surface with 4 nodes.

* Clebsch surface Rational. The surface Σ"x"_{"i"}= Σ"x"_{"i"}^{3}= 0 in "P"^{4}.

*cubic surface s Rational.

*Del Pezzo surface s Rational. Anticanonical divisor is ample, for example "P"^{2}blown up in at most 8 points.

* Dolgachev surfaces Elliptic.

*elliptic surface s Surfaces with an elliptic fibration.

*Enriques surface s (κ = 0)

* exceptional surfaces: Picard number has the maximal possible value "h"^{1,1}.

* fake projective plane general type, found by Mumford, same betti numbers as projective plane.

* Fano surfaces Rational. Same as del Pezzo surfaces.

* Fermat surface of degree "d": Solutions of "w"^{"d"}+ "x"^{"d"}+ "y"^{"d"}+ "z"^{"d"}= 0 in "P"^{3}.

* general type κ = 2

* Godeaux surfaces (general type)

*Hilbert modular surface s

* Hirzebruch surfaces Rational ruled surfaces.

* Hopf surfaces κ = −∞, non-algebraic, class VII

* Horikawa surfaces general type

* Horrocks-Mumford surfaces. These are certain abelian surfaces of degree 10 in "P"^{4}, given as zero sets of sections of the rank 2Horrocks-Mumford bundle .

*Humbert surface s These are certain surfaces in quotients of the Siegel upper half plane of genus 2.

* hyperelliptic surfaces κ = 0, same as bielliptic surfaces.

* Inoue surfaces κ = −∞, class VII,"b"_{2}= 0. (Several quite different families were also found by Inoue, and are also sometimes called Inoue surfaces.)

*Inoue-Hirzebruch surface s κ = −∞, non-algebraic, type VII, "b"_{2}>0.

*K3 surface s κ = 0,supersingular K3 surface .

* Kähler surfaces complex surfaces with a Kähler metric, which exists if and only if the first betti number "b"_{1}is even.

* Kodaira surfaces κ = 0, non-algebraic

*Kummer surface s κ = 0, special sorts of K3 surfaces.

* minimal surfaces Surfaces with no rational −1 curves. (They have no connection with minimal surfaces in differential geometry.)

* Mumford surface A "fake projective plane"

* non-classical Enriques surface Only in characteristic 2.

* numerical Campedelli surfaces surfaces of general type with the same Hodge numbers as a Campedelli surface.

* numerical Godeaux surfaces surfaces of general type with the same Hodge numbers as a Godeaux surface.

*projective plane Rational

* properly elliptic surfaces κ = 1, elliptic surfaces of genus ≥2.

* quadric surfaces Rational, isomorphic to "P"^{1}×"P"^{1}.

* quartic surfaces Nonsingular ones are K3s.

* quasi Enriques surface These only exist in characteristic 2.

* quasi elliptic surface Only in characteristic "p">0.

* quotient surfaces: Quotients of surfaces by finite groups. Examples: Kummer, Godeaux, Hopf, Inoue surfaces.

*rational surface s κ = −∞, birational to projective plane

*ruled surface s κ = −∞

* [*http://enriques.mathematik.uni-mainz.de/docs/Esarti.shtml Sarti surface*] A degree-12 surface in "P"^{3}with 600 nodes.

*Steiner surface A surface in "P"^{4}with singularities which is birational to the projective plane.

*surface of general type κ = 2.

*Togliatti surface s [*http://enriques.mathematik.uni-mainz.de/docs/Etogliatti.shtml*] , degree-5 surfaces in "P"^{3}with 31 nodes.

* unirational surfaces Castelnuovo proved these are all rational in characteristic 0.

*Veronese surface An embedding of the projective plane into "P"^{5}.

* Weddle surface κ = 0, birational to Kummer surface.

*Zariski surface s (only in characteristic "p" > 0): There is a purely inseparable dominant rational map of degree "p" from the projective plane to the surface.**ee also***

Enriques-Kodaira classification

*Algebraic surface

*List of surfaces **References*** "Compact Complex Surfaces" by Wolf P. Barth, Klaus Hulek, Chris A.M. Peters, Antonius Van de Ven ISBN 3-540-00832-2

* "Complex algebraic surfaces" by Arnaud Beauville, ISBN 0521288150**External links*** Mathworld has a long list of [

*http://mathworld.wolfram.com/topics/AlgebraicSurfaces.html algebraic surfaces*] with pictures.

* Some more [*http://www.AlgebraicSurface.net pictures of algebraic surfaces*] , especially ones with many nodes.

*Wikimedia Foundation.
2010.*

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