# List of complex and algebraic surfaces

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

## 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 $_{n+1}C_{2}$ points by the linear system of degree-$n$ curves through those points
** Bordiga surfaces, the White surfaces determined by families of quartic curves

### 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 $\mathbb {P} ^{3}$.
* 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 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