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.

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

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 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