# List of complex and algebraic surfaces

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

## Contents

## Kodaira dimension [edit]

### Rational surfaces[edit]

#### Quadric surfaces[edit]

#### Rational cubic surfaces[edit]

- 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[edit]

- 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[edit]

- 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[edit]

- 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 points by the linear system of degree- curves through those points
- Bordiga surfaces, the White surfaces determined by families of quartic curves

### Non-rational ruled surfaces[edit]

### Class VII surfaces[edit]

- Vanishing second Betti number:
- Hopf surfaces
- Inoue surfaces; several other families discovered by Inoue have also been called "Inoue surfaces"

- Positive second Betti number:

## Kodaira dimension [edit]

### K3 surfaces[edit]

- 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[edit]

- Reye congruences, the locus of lines that lie on two out of three general quadric surfaces in projective space

### Abelian surfaces[edit]

*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- surfaces[edit]

- 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 [edit]

## Kodaira dimension (surfaces of general type)[edit]

- 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[edit]

- Surfaces that are also Shimura varieties:
- 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
*h*^{1,1} - Kähler surfaces, complex surfaces with a Kähler metric; equivalently, surfaces for which the first Betti number
*b*_{1}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[edit]

## References[edit]

*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[edit]

- 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.