This is a list of named (classes of) algebraic surfaces
and complex surfaces
. The notation κ stands for the Kodaira dimension
, which divides surfaces into four coarse classes.
Algebraic and complex surfaces
- abelian surfaces (κ = 0) Two dimensional abelian varieties.
- algebraic surfaces
- Barlow surfaces General type, simply connected.
- Barth sextic A degree-6 surface in P3 with 65 nodes.
- Barth decic A degree-10 surface in P3 with 345 nodes.
- Beauville surfaces General type
- bielliptic surfaces (κ = 0) Same as hyperelliptic surfaces.
- Bordiga surfaces A degree-6 embedding of the projective plane into P4 defined by the quartics through 10 points in general position.
- Burniat surfaces General type
- Campedelli surfaces General type
- Castelnuovo surfaces General type
- Catanese surfaces General type
- class VII surfaces κ = −∞, non-algebraic.
- Cayley surface Rational. A cubic surface with 4 nodes.
- Clebsch surface Rational. The surface Σxi = Σxi3 = 0 in P4.
- cubic surfaces Rational.
- Del Pezzo surfaces Rational. Anticanonical divisor is ample, for example P2 blown up in at most 8 points.
- Dolgachev surfaces Elliptic.
- elliptic surfaces Surfaces with an elliptic fibration.
- Enriques surfaces (κ = 0)
- exceptional surfaces: Picard number has the maximal possible value h1,1.
- fake projective plane general type, found by Mumford, same betti numbers as projective plane.
- Fano surfaces Rational. Same as del Pezzo surfaces.
- Fermat surface of degree d: Solutions of wd + xd + yd + zd = 0 in P3.
- general type κ = 2
- Godeaux surfaces (general type)
- Hilbert modular surfaces
- Hirzebruch surfaces Rational ruled surfaces.
- Hopf surfaces κ = −∞, non-algebraic, class VII
- Horikawa surfaces general type
- Horrocks-Mumford surfaces. These are certain abelian surfaces of degree 10 in P4, given as zero sets of sections of the rank 2 Horrocks-Mumford bundle.
- Humbert surfaces These are certain surfaces in quotients of the Siegel upper half plane of genus 2.
- hyperelliptic surfaces κ = 0, same as bielliptic surfaces.
- Inoue surfaces κ = −∞, class VII,b2 = 0. (Several quite different families were also found by Inoue, and are also sometimes called Inoue surfaces.)
- Inoue-Hirzebruch surfaces κ = −∞, non-algebraic, type VII, b2>0.
- K3 surfaces κ = 0, supersingular K3 surface.
- Kähler surfaces complex surfaces with a Kähler metric, which exists if and only if the first betti number b1 is even.
- Kodaira surfaces κ = 0, non-algebraic
- Kummer surfaces κ = 0, special sorts of K3 surfaces.
- minimal surfaces Surfaces with no rational −1 curves. (They have no connection with minimal surfaces in differential geometry.)
- Mumford surface A "fake projective plane"
- non-classical Enriques surface Only in characteristic 2.
- numerical Campedelli surfaces surfaces of general type with the same Hodge numbers as a Campedelli surface.
- numerical Godeaux surfaces surfaces of general type with the same Hodge numbers as a Godeaux surface.
- projective plane Rational
- properly elliptic surfaces κ = 1, elliptic surfaces of genus ≥2.
- quadric surfaces Rational, isomorphic to P1×P1.
- quartic surfaces Nonsingular ones are K3s.
- quasi Enriques surface These only exist in characteristic 2.
- quasi elliptic surface Only in characteristic p>0.
- quotient surfaces: Quotients of surfaces by finite groups. Examples: Kummer, Godeaux, Hopf, Inoue surfaces.
- rational surfaces κ = −∞, birational to projective plane
- ruled surfaces κ = −∞
- Sarti surface A degree-12 surface in P3 with 600 nodes.
- Steiner surface A surface in P4 with singularities which is birational to the projective plane.
- surface of general type κ = 2.
- Togliatti surfaces , degree-5 surfaces in P3 with 31 nodes.
- unirational surfaces Castelnuovo proved these are all rational in characteristic 0.
- Veronese surface An embedding of the projective plane into P5.
- Weddle surface κ = 0, birational to Kummer surface.
- Zariski surfaces (only in characteristic p > 0): There is a purely inseparable dominant rational map of degree p from the projective plane to the surface.
- 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 0521288150