List of complex and 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 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
• Cayley nodal cubic surface Rational. A cubic surface with 4 nodes.
• Cayley's ruled cubic surface
• Châtelet surfaces Rational
• class VII surfaces κ = −∞, non-algebraic.
• Clebsch surface Rational. The surface Σx_i = Σx_i³ = 0 in P⁴.
• Coble surfaces Rational
• 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.
• Endrass surface A surface of degree 8 with 168 nodes
• Enneper surface
• Enoki surface Class VII
• 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 surface of lines on a non-singular 3-fold. It can also mean del Pezzo surface.
• Fermat surface of degree d: Solutions of w^d + x^d + y^d + z^d = 0 in P³.
• general type κ = 2
• generalized Raynaud surface in positive characteristic
• 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-space 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.
• Kato surface Class VII
• Klein icosahedral surface The Clebsch cubic surface or its blowup in 10 points.
• Kodaira surfaces κ = 0, non-algebraic
• Kummer surfaces κ = 0, special sorts of K3 surfaces.
• Labs surface A surface of degree 7 with 99 nodes
• 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.
• Picard modular surface
• Plücker surface Birational to Kummer 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.
• quasi-hyperelliptic surface
• quotient surfaces: Quotients of surfaces by finite groups. Examples: Kummer, Godeaux, Hopf, Inoue surfaces.
• rational surfaces κ = −∞, birational to projective plane
• Raynaud surface in positive characteristic
• Reye congruence A special sort of Enriques surface. κ=0.
• Roman surface
• ruled surfaces κ = −∞
• Sarti surface A degree-12 surface in P³ with 600 nodes.
• Segre surface An intersection of two quadrics, isomorphic to the projective plane blown up in 5 points.
• Steiner surface A surface in P⁴ with singularities which is birational to the projective plane.
• surface of general type κ = 2.
• Tetrahedroid A special Kummer surface.
• Togliatti surfaces, degree-5 surfaces in P³ with 31 nodes.
• Todorov surface. General type.
• unirational surfaces Castelnuovo proved these are all rational in characteristic 0.
• Veronese surface An embedding of the projective plane into P⁵.
• Wave surface A special Kummer surface.
• Weddle surface κ = 0, birational to Kummer surface.
• White surface Rational.
• 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.

See also

• Enriques–Kodaira classification
• List of surfaces

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

External links

• Mathworld has a long list of algebraic surfaces with pictures.
• Some more pictures of algebraic surfaces, especially ones with many nodes.
• Pictures of algebraic surfaces by Herwig Hauser.
• Free program SURFER to visualize algebraic surfaces in real-time, including a user gallery.