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.


Wikimedia Foundation. 2010.

Игры ⚽ Нужна курсовая?

Look at other dictionaries:

  • List of algebraic geometry topics — This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 …   Wikipedia

  • Algebraic surface — In mathematics, an algebraic surface is an algebraic 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 a complex manifold, when it is non singular)… …   Wikipedia

  • List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… …   Wikipedia

  • Algebraic geometry — This Togliatti surface is an algebraic surface of degree five. Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It… …   Wikipedia

  • List of surfaces — This is a list of surfaces, by Wikipedia page. See also list of curves, Riemann surface. Minimal surfaces* Costa surface * Catenoid * Riemann s surface * Saddle tower * Gyroid * Catalan surface * Enneper surface =Non orientable surfaces= *Klein… …   Wikipedia

  • List of mathematical examples — This page will attempt to list examples in mathematics. To qualify for inclusion, an article should be about a mathematical object with a fair amount of concreteness. Usually a definition of an abstract concept, a theorem, or a proof would not be …   Wikipedia

  • List of geometric topology topics — This is a list of geometric topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of algebraic topology topics Publications in topology Contents 1 Low dimensional topology 1.1 …   Wikipedia

  • List of important publications in mathematics — One of the oldest surviving fragments of Euclid s Elements, found at Oxyrhynchus and dated to circa AD 100. The diagram accompanies Book II, Proposition 5.[1] This is a list of important publications in mathematics, organized by field. Some… …   Wikipedia

  • Algebraic curve — In algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with… …   Wikipedia

  • List of theorems — This is a list of theorems, by Wikipedia page. See also *list of fundamental theorems *list of lemmas *list of conjectures *list of inequalities *list of mathematical proofs *list of misnamed theorems *Existence theorem *Classification of finite… …   Wikipedia

Share the article and excerpts

Direct link
… Do a right-click on the link above
and select “Copy Link”