List of complex and algebraic surfaces

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This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification.

## Contents

* 1 Kodaira dimension −∞
* 1.1 Rational surfaces
* 1.1.1 Quadric surfaces
* 1.1.2 Rational cubic surfaces
* 1.1.3 Rational quartic surfaces
* 1.1.4 Other rational surfaces in space
* 1.1.5 Other families of rational surfaces
* 1.2 Non-rational ruled surfaces
* 1.3 Class VII surfaces
* 2 Kodaira dimension 0
* 2.1 K3 surfaces
* 2.2 Enriques surfaces
* 2.3 Abelian surfaces
* 2.4 Other classes of dimension-0 surfaces
* 3 Kodaira dimension 1
* 4 Kodaira dimension 2 (surfaces of general type)
* 5 Families of surfaces with members in multiple classes
* 6 See also
* 7 References
* 8 External links

## Kodaira dimension −∞

### Rational surfaces

* Projective plane

#### Quadric surfaces

* Cone (geometry)
* Cylinder
* Ellipsoid
* Hyperboloid
* Paraboloid
* Sphere
* Spheroid

#### Rational cubic surfaces

* Cayley nodal cubic surface, a certain cubic surface with 4 nodes
* Cayley's ruled cubic surface
* Clebsch surface or Klein icosahedral surface
* Fermat cubic
* Monkey saddle
* Parabolic conoid
* Plücker's conoid
* Whitney umbrella

#### Rational quartic surfaces

* Châtelet surfaces
* Dupin cyclides, inversions of a cylinder, torus, or double cone in a sphere
* Gabriel's horn
* Right circular conoid
* Roman surface or Steiner surface, a realization of the real projective plane in real affine space
* Tori, surfaces of revolution generated by a circle about a coplanar axis

#### Other rational surfaces in space

* Boy's surface, a sextic realization of the real projective plane in real affine space
* Enneper surface, a nonic minimal surface
* Henneberg surface, a minimal surface of degree 15
* Bour's minimal surface, a surface of degree 16
* Richmond surfaces, a family of minimal surfaces of variable degree

#### Other families of rational surfaces

* Coble surfaces
* Del Pezzo surfaces, surfaces with an ample anticanonical divisor
* Hirzebruch surfaces, rational ruled surfaces
* Segre surfaces, intersections of two quadrics in projective 4-space
* Unirational surfaces of characteristic 0
* Veronese surface, the Veronese embedding of the projective plane into projective 5-space
* White surfaces, the blow-up of the projective plane at n+1C2 points by the linear system of degree-n curves through those points
* Bordiga surfaces, the White surfaces determined by families of quartic curves

### Non-rational ruled surfaces

### Class VII surfaces

* Vanishing second Betti number:
* Hopf surfaces
* Inoue surfaces; several other families discovered by Inoue have also been called "Inoue surfaces"
* Positive second Betti number:
* Enoki surfaces
* Inoue–Hirzebruch surfaces
* Kato surfaces

## Kodaira dimension 0

### K3 surfaces

* Kummer surfaces
* Tetrahedroids, special Kummer surfaces
* Wave surface, a special tetrahedroid
* Plücker surfaces, birational to Kummer surfaces
* Weddle surfaces, birational to Kummer surfaces
* Smooth quartic surfaces
* Supersingular K3 surfaces

### Enriques surfaces

* Reye congruences, the locus of lines that lie on at least two quadrics in a general three dimensional linear system of quadric surfaces in projective 3-space ℙ3.
* The quotient of a K3 surface under a fixpointfree involution.

### Abelian surfaces

* Horrocks–Mumford surfaces, surfaces of degree 10 in projective 4-space that are the zero locus of sections of the rank-two Horrocks–Mumford bundle

### Other classes of dimension-0 surfaces

* Non-classical Enriques surfaces, a variation on the notion of Enriques surfaces that only exist in characteristic two
* Hyperelliptic surfaces or bielliptic surfaces; quasi-hyperelliptic surfaces are a variation of this notion that exist only in characteristics two and three
* Kodaira surfaces

## Kodaira dimension 1

* Dolgachev surfaces

## Kodaira dimension 2 (surfaces of general type)

* Barlow surfaces
* Beauville surfaces
* Burniat surfaces
* Campedelli surfaces; surfaces of general type with the same Hodge numbers as Campedelli surfaces are called numerical Campidelli surfaces
* Castelnuovo surfaces
* Catanese surfaces
* Fake projective planes or Mumford surfaces, surfaces with the same Betti numbers as projective plane but not isomorphic to it
* Fano surface of lines on a non-singular 3-fold; sometimes, this term is taken to mean del Pezzo surface
* Godeaux surfaces; surfaces of general type with the same Hodge numbers as Godeaux surfaces are called numerical Godeaux surfaces
* Horikawa surfaces
* Todorov surfaces

## Families of surfaces with members in multiple classes

* Surfaces that are also Shimura varieties:
* Hilbert modular surfaces
* Humbert surfaces
* Picard modular surfaces
* Shioda modular surfaces
* Elliptic surfaces, surfaces with an elliptic fibration; quasielliptic surfaces constitute a modification this idea that occurs in finite characteristic
* Raynaud surfaces and generalized Raynaud surfaces, certain quasielliptic counterexamples to the conclusions of the Kodaira vanishing theorem
* Exceptional surfaces, surfaces whose Picard number achieve the bound set by the central Hodge number h1,1
* Kähler surfaces, complex surfaces with a Kähler metric; equivalently, surfaces for which the first Betti number b1 is even
* Minimal surfaces, surfaces that can't be obtained from another by blowing up at a point; they have no connection with the minimal surfaces of differential geometry
* Nodal surfaces, surfaces whose only singularities are nodes
* Cayley's nodal cubic, which has 4 nodes
* Kummer surfaces, quartic surfaces with 16 nodes
* Togliatti surface, a certain quintic with 31 nodes
* Barth surfaces, referring to a certain sextic with 65 nodes and decic with 345 nodes
* Labs surface, a certain septic with 99 nodes
* Endrass surface, a certain surface of degree 8 with 168 nodes
* Sarti surface, a certain surface of degree 12 with 600 nodes
* Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue surfaces
* Zariski surfaces, surfaces in finite characteristic that admit a purely inseparable dominant rational map from the projective plane

## See also

* Enriques–Kodaira classification
* 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 0-521-28815-0

## 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.
* Free program SURFER to visualize algebraic surfaces in real-time, including a user gallery.

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