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 the Enriques–Kodaira classification.

Kodaira dimension [edit]

Rational surfaces[edit]

Quadric surfaces[edit]

Rational cubic surfaces[edit]

Rational quartic surfaces[edit]

Other rational surfaces in space[edit]

Other families of rational surfaces[edit]

Non-rational ruled surfaces[edit]

Class VII surfaces[edit]

Kodaira dimension [edit]

K3 surfaces[edit]

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]

Kodaira dimension [edit]

Kodaira dimension (surfaces of general type)[edit]

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
  • 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[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]