Algebraic type

List of algebraic surfaces

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 Σx_i = Σx_i³ = 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 h^1,1.
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 w^d + x^d + y^d + z^d = 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.

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 0521288150