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 Σxi = Σxi3 = 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 h1,1.
- 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 wd + xd + yd + zd = 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
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.
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