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.

Kodaira dimension

Rational surfaces

Quadric surfaces

Rational cubic surfaces

Rational quartic surfaces

Other rational surfaces in space

Other families of rational surfaces

Class VII surfaces

Kodaira dimension

K3 surfaces

Enriques surfaces

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

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- surfaces

Kodaira dimension

Kodaira dimension (surfaces of general type)

Families of surfaces with members in multiple classes

  • 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

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
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