List of algebraic surfaces

List of algebraic surfaces

This is a list of named (classes of) algebraic surfaces and complex surfaces. The notation κ stands for the Kodaira dimension, which divides surfaces into four coarse classes.

Algebraic and complex surfaces

* abelian surfaces (κ = 0) Two dimensional abelian varieties.
* algebraic surfaces
* 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 surfaces Rational.
* Del Pezzo surfaces Rational. Anticanonical divisor is ample, for example "P"2 blown up in at most 8 points.
* Dolgachev surfaces Elliptic.
* elliptic surfaces Surfaces with an elliptic fibration.
* Enriques surfaces (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 surfaces
* 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 2 Horrocks-Mumford bundle.
* Humbert surfaces 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 surfaces κ = −∞, non-algebraic, type VII, "b"2>0.
* K3 surfaces 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 surfaces 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 surfaces κ = −∞, birational to projective plane
* ruled surfaces 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 surfaces [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 surfaces (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.


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