Algebraic and complex surfaces

• abelian surface
  Abelian surface
  In mathematics, an abelian surface is 2-dimensional abelian variety.One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic...
  
  s (κ = 0) Two-dimensional abelian varieties.
• algebraic surface
  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 has complex dimension two and so of dimension four as a smooth manifold.The theory of algebraic surfaces is much more complicated than that...
  
  s
• Barlow surface
  Barlow surface
  In mathematics, a Barlow surface is one of the complex surfaces introduced by .They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points....
  
  s General type, simply connected.
• Barth surface
  Barth surface
  In algebraic geometry, a Barth surface is one of the complex surfaces in 3 dimensions with large numbers of double points found by . Two examples are the Barth sextic of degree 6 with 65 double points, and the Barth decic of degree 10 with 345 double points.Some admit icosahedral symmetry.-External...
  
  s Surfaces of degrees 6 and 10 with many nodes.
• Beauville surface
  Beauville surface
  In mathematics, a Beauville surface is one of the surfaces of general type introduced by . They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.-Construction:...
  
  s General type
• bielliptic surfaces (κ = 0) Same as hyperelliptic surfaces.
• Bordiga surface
  Bordiga surface
  In algebraic geometry, a Bordiga surface is a certain sort of rational surface of degree 6 in P4.A Bordiga surface is isomorphic to the projective plane blown up in 10 points, the embedding into P4 is given by the 5-dimensional space of quartics passing through the 10 points. White surfaces...
  
  s A degree-6 embedding of the projective plane into P⁴ defined by the quartics through 10 points in general position.
• Burniat surface
  Burniat surface
  In mathematics, a Burniat surface is one of the surfaces of general type introduced by .-Invariants:The geometric genus and irregularity are both 0. The Chern number c12 is 2, 3, 4, 5, or 6....
  
  s General type
• Campedelli surface
  Campedelli surface
  In mathematics, a Campedelli surface is one of the surfaces of general type introduced by Campedelli.Surfaces with the same Hodge numbers are called numerical Campedelli surfaces.-Invariants:Hodge diamond:10 0080001...
  
  s General type
• Castelnuovo surface
  Castelnuovo surface
  In mathematics, a Castelnuovo surface is a surface of general type such that the canonical bundle is very ample andsuch that c12 = 3pg − 7. Guido Castelnuovo proved that if the canonical bundle is very ample for a surface of general type then...
  
  s General type
• Catanese surface
  Catanese surface
  In mathematics, a Catanese surface is one of the surfaces of general type introduced by .-Construction:The construction starts with a quintic V with 20 double points. Let W be the surface obtained by blowing up the 20 double points. Suppose that W has a double cover X branched over the 20...
  
  s General type
• Cayley surface
  Cayley surface
  Cayley surface may refer to:*Cayley's nodal cubic surface*Cayley's ruled cubic surface...
  
   Rational. A cubic surface with 4 nodes.
• Châtelet surface
  Châtelet surface
  In algebraic geometry, a Châtelet surface is a rational surface studied by given by an equationy^2-az^2=P \, where P has degree 3 or 4. They are conic bundles....
  
  s Rational
• class VII surfaces κ = −∞, non-algebraic.
• Clebsch surface Rational. The surface Σx_i = Σx_i³ = 0 in P⁴.
• Coble surface
  Coble surface
  In algebraic geometry, a Coble surface was defined by to be a smooth rational projective surface with empty anti-canonical linear system |−K| and non-empty anti-bicanonical linear system |−2K|. An example of a Coble surface is the blowing up of the projective plane at the 10 nodes of...
  
  s Rational
• cubic surface
  Cubic surface
  A cubic surface is a projective variety studied in algebraic geometry. It is an algebraic surface in three-dimensional projective space defined by a single polynomial which is homogeneous of degree 3...
  
  s Rational.
• Del Pezzo surface
  Del Pezzo surface
  In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class...
  
  s Rational. Anticanonical divisor is ample, for example P² blown up in at most 8 points.
• Dolgachev surface
  Dolgachev surface
  In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds no two of which are diffeomorphic.-Properties:...
  
  s Elliptic.
• elliptic surface
  Elliptic surface
  In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
  
  s Surfaces with an elliptic fibration.
• Enoki surface
  Enoki surface
  In mathematics, an Enoki surface is compact complex surface with positive second Betti number that has a global spherical shell and a non-trivial divisor D with H0 ≠ 0 and  = 0. constructed some examples. They are surfaces of class VII, so are non-Kähler and have Kodaira...
  
   Class VII
• Enriques surface
  Enriques surface
  In mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
  
  s (κ = 0)
• exceptional surfaces: Picard number has the maximal possible value h^1,1.
• fake projective plane
  Fake projective plane
  In mathematics, a fake projective plane is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such objects are always algebraic surfaces of general type....
  
   general type, found by Mumford
  David Mumford
  David Bryant Mumford is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science...
  
  , same Betti number
  Betti number
  In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
  
  s as projective plane.
• Fano surface
  Fano surface
  In algebraic geometry, a Fano surface is a surface of general type whose points index the lines on a non-singular cubic threefold...
  
   of lines on a non-singular 3-fold. It can also mean del Pezzo surface.
• Fermat surface of degree d: Solutions of w^d + x^d + y^d + z^d = 0 in P³.
• general type
  Surface of general type
  In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this...
  
   κ = 2
• generalized Raynaud surface in positive characteristic
• Godeaux surface
  Godeaux surface
  In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux.Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes called Godeaux surfaces...
  
  s (general type)
• Hilbert modular surface
  Hilbert modular surface
  In mathematics, a Hilbert modular surface is one of the surfaces obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular group....
  
  s
• Hirzebruch surface
  Hirzebruch surface
  In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by .-Definition:The Hirzebruch surfaceΣn is the P1 bundle over P1associated to the sheafO+O.\...
  
  s Rational ruled surfaces.
• Hopf surface
  Hopf surface
  In complex geometry, a Hopf surface is a compact complex surface obtainedas a quotient of the complex vector space C2 \ 0...
  
  s κ = −∞, non-algebraic, class VII
• Horikawa surface
  Horikawa surface
  In mathematics, a Horikawa surface is one of the surfaces of general type introduced by Horikawa.These are surfaces with q = 0 and pg = c12/2 + 2 or c12/2 + 3/2 .They are all simply connected, and Horikawa gave a...
  
  s general type
• Horrocks–Mumford surface
  Abelian surface
  In mathematics, an abelian surface is 2-dimensional abelian variety.One dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dimension 2 are not algebraic...
  
  s. These are certain abelian surfaces of degree 10 in P⁴, 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-space of genus 2.
• hyperelliptic surface
  Hyperelliptic surface
  In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface with an elliptic fibration over an elliptic curve. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group....
  
  s κ = 0, same as bielliptic surfaces.
• Inoue surface
  Inoue surface
  In complex geometry, a part of mathematics, the termInoue surface denotes several complex surfacesof Kodaira class VII. They arenamed after Masahisa Inoue, who gave the first non-trivialexamples of Kodaira class VII surfaces in 1974....
  
  s κ = −∞, class VII,b₂ = 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, b₂>0.
• K3 surface
  K3 surface
  In mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
  
  s κ = 0, supersingular K3 surface
  Supersingular K3 surface
  In algebraic geometry, a supersingular K3 surface is a K3 surface whose l-adic cohomology is generated by algebraic cycles; in other words, since the second Betti number of a K3 surface is always 22, such a surface must possess 22 independent elements in its Picard group .Such surfaces can exist...
  
  .
• Kähler surface
  Kähler manifold
  In mathematics, a Kähler manifold is a manifold with unitary structure satisfying an integrability condition.In particular, it is a Riemannian manifold, a complex manifold, and a symplectic manifold, with these three structures all mutually compatible.This threefold structure corresponds to the...
  
  s complex surfaces with a Kähler metric, which exists if and only if the first Betti number b₁ is even.
• Kato surface
  Kato surface
  In mathematics, a Kato surface is a compact complex surface with positive first Betti number that has a global spherical shell. showed that Kato surfaces have small analytic deformations that are the blowups of primary Hopf surfaces at a finite number of points. In particular they have an infinite...
  
   Class VII
• Klein icosahedral surface The Clebsch cubic surface or its blowup in 10 points.
• Kodaira surface
  Kodaira surface
  In mathematics, a Kodaira surface is a compact complex surface of Kodaira dimension 0 and odd first Betti number.These are never algebraic, though they have non-constant meromorphic functions...
  
  s κ = 0, non-algebraic
• Kummer surfaces κ = 0, special sorts of K3 surfaces.
• minimal surface
  Enriques-Kodaira classification
  In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a moduli space...
  
  s 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
  Enriques surface
  In mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
  
   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.
• Plücker surface
  Plücker surface
  In algebraic geometry, a Plücker surface, studied by , is a quartic surface in 3-dimensional projective space with a double line and 8 nodes.-Construction:For any quadric line complex, the lines of the complex in a plane envelop a quadric in the plane...
  
   Birational to Kummer surface
• projective plane
  Projective plane
  In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines that do not intersect...
  
   Rational
• properly elliptic surface
  Elliptic surface
  In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
  
  s κ = 1, elliptic surfaces of genus ≥2.
• quadric surface
  Quadric
  In mathematics, a quadric, or quadric surface, is any D-dimensional hypersurface in -dimensional space defined as the locus of zeros of a quadratic polynomial...
  
  s Rational, isomorphic to P¹ × P¹.
• quartic surface
  Quartic surface
  In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4.More specifically there are two closely related types of quartic surface: affine and projective...
  
  s Nonsingular ones are K3s.
• quasi Enriques surface
  Enriques surface
  In mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
  
   These only exist in characteristic 2.
• quasi elliptic surface
  Elliptic surface
  In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper connected morphism to an algebraic curve, almost all of whose fibers are elliptic curves....
  
   Only in characteristic p > 0.
• quasi-hyperelliptic surface
• quotient surfaces: Quotients of surfaces by finite groups. Examples: Kummer, Godeaux, Hopf, Inoue surfaces.
• rational surface
  Rational surface
  In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two...
  
  s κ = −∞, birational to projective plane
• Raynaud surface
  Raynaud surface
  In mathematics, a Raynaud surface is a particular kind of algebraic surface that was introduced in and named for . To be precise, a Raynaud surface is a quasi-elliptic surface over an algebraic curve of genus g greater than 1, such that all fibers are irreducible and the fibration has a section...
  
   in positive characteristic
• Roman surface
  Roman surface
  The Roman surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry...
  
  
• ruled surfaces κ = −∞
• Sarti surface A degree-12 surface in P³ with 600 nodes.
• Segre surface
  Segre surface
  In algebraic geometry, a Segre surface, named for Beniamino Segre, is an intersection of two quadrics in 4-dimensional projective space.They are rational surfaces isomorphic to a projective plane blown up in 5 points with no 3 on a line, and are del Pezzo surfaces of degree 4, and have 16 rational...
  
   An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points.
• Steiner surface
  Steiner surface
  In geometry, a branch of mathematics, the Steiner surfaces, discovered by Jakob Steiner, are mappings of the real projective plane into three-dimensional real projective space...
  
   A surface in P⁴ with singularities which is birational to the projective plane.
• surface of general type
  Surface of general type
  In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 will actually be an algebraic surface, and in some sense most surfaces are in this...
  
   κ = 2.
• Togliatti surfaces, degree-5 surfaces in P³ with 31 nodes.
• unirational surface
  Rational surface
  In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two...
  
  s Castelnuovo proved these are all rational in characteristic 0.
• Veronese surface
  Veronese surface
  In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese...
  
   An embedding of the projective plane into P⁵.
• Weddle surface
  Weddle surface
  In algebraic geometry, a Weddle surface is a quartic surface in 3-dimensional projective space, given by the locus of vertices of the 3-dimensional family of cones passing through 6 points in general position, and studied by ....
  
   κ = 0, birational to Kummer surface.
• White surface
  White surface
  In algebraic geometry, a White surface is one of the rational surfaces in Pn studied by , generalizing cubic surfaces and Bordiga surfaces, which are the cases n = 3 or 4....
  
   Rational.
• Zariski surface
  Zariski surface
  In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane to the 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.

External links

• Mathworld has a long list of algebraic surfaces with pictures.
• Some more pictures of algebraic surfaces, especially ones with many nodes.
• Pictures of algebraic surfaces by Herwig Hauser.