List of algebraic surfaces

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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.

[edit] Algebraic and complex surfaces

• abelian surfaces (κ = 0) Two dimensional abelian varieties.
• algebraic surfaces
• Barlow surfaces General type, simply connected.
• Barth surfaces Surfaces of degrees 6 and 10 with many 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.

[edit] See also

• Enriques-Kodaira classification
• Algebraic surface
• List of surfaces

[edit] 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

[edit] External links

• Mathworld has a long list of algebraic surfaces with pictures.
• Some more pictures of algebraic surfaces, especially ones with many nodes.