• Source: List of complex and algebraic surfaces

This is a list of named algebraic surfaces, compact complex surfaces, and families thereof, sorted according to their Kodaira dimension following Enriques–Kodaira classification.




Kodaira dimension −∞






= Rational surfaces

=
Projective plane



Quadric surfaces

Cone (geometry)
Cylinder
Ellipsoid
Hyperboloid
Paraboloid
Sphere
Spheroid



Rational cubic surfaces

Cayley nodal cubic surface, a certain cubic surface with 4 nodes
Cayley's ruled cubic surface
Clebsch surface or Klein icosahedral surface
Fermat cubic
Monkey saddle
Parabolic conoid
Plücker's conoid
Whitney umbrella



Rational quartic surfaces

Châtelet surfaces
Dupin cyclides, inversions of a cylinder, torus, or double cone in a sphere
Gabriel's horn
Right circular conoid
Roman surface or Steiner surface, a realization of the real projective plane in real affine space
Tori, surfaces of revolution generated by a circle about a coplanar axis



Other rational surfaces in space

Boy's surface, a sextic realization of the real projective plane in real affine space
Enneper surface, a nonic minimal surface
Henneberg surface, a minimal surface of degree 15
Bour's minimal surface, a surface of degree 16
Richmond surfaces, a family of minimal surfaces of variable degree



Other families of rational surfaces

Coble surfaces
Del Pezzo surfaces, surfaces with an ample anticanonical divisor
Hirzebruch surfaces, rational ruled surfaces
Segre surfaces, intersections of two quadrics in projective 4-space
Unirational surfaces of characteristic 0
Veronese surface, the Veronese embedding of the projective plane into projective 5-space
White surfaces, the blow-up of the projective plane at






n
+
1



C

2




{\displaystyle _{n+1}C_{2}}

points by the linear system of degree-



n


{\displaystyle n}

curves through those points
Bordiga surfaces, the White surfaces determined by families of quartic curves




= Non-rational ruled surfaces

=




= Class VII surfaces

=
Vanishing second Betti number:
Hopf surfaces
Inoue surfaces; several other families discovered by Inoue have also been called "Inoue surfaces"
Positive second Betti number:
Enoki surfaces
Inoue–Hirzebruch surfaces
Kato surfaces




Kodaira dimension 0






= K3 surfaces

=
Kummer surfaces
Tetrahedroids, special Kummer surfaces
Wave surface, a special tetrahedroid
Plücker surfaces, birational to Kummer surfaces
Weddle surfaces, birational to Kummer surfaces
Smooth quartic surfaces
Supersingular K3 surfaces




= Enriques surfaces

=
Reye congruences, the locus of lines that lie on at least two quadrics in a general three dimensional linear system of quadric surfaces in projective 3-space





P


3




{\displaystyle \mathbb {P} ^{3}}

.
The quotient of a K3 surface under a fixpointfree involution.




= Abelian surfaces

=
Horrocks–Mumford surfaces, surfaces of degree 10 in projective 4-space that are the zero locus of sections of the rank-two Horrocks–Mumford bundle




= Other classes of dimension-0 surfaces

=
Non-classical Enriques surfaces, a variation on the notion of Enriques surfaces that only exist in characteristic two
Hyperelliptic surfaces or bielliptic surfaces; quasi-hyperelliptic surfaces are a variation of this notion that exist only in characteristics two and three
Kodaira surfaces




Kodaira dimension 1


Dolgachev surfaces




Kodaira dimension 2 (surfaces of general type)


Barlow surfaces
Beauville surfaces
Burniat surfaces
Campedelli surfaces; surfaces of general type with the same Hodge numbers as Campedelli surfaces are called numerical Campidelli surfaces
Castelnuovo surfaces
Catanese surfaces
Fake projective planes or Mumford surfaces, surfaces with the same Betti numbers as projective plane but not isomorphic to it
Fano surface of lines on a non-singular 3-fold; sometimes, this term is taken to mean del Pezzo surface
Godeaux surfaces; surfaces of general type with the same Hodge numbers as Godeaux surfaces are called numerical Godeaux surfaces
Horikawa surfaces
Todorov surfaces




Families of surfaces with members in multiple classes


Surfaces that are also Shimura varieties:
Hilbert modular surfaces
Humbert surfaces
Picard modular surfaces
Shioda modular surfaces
Elliptic surfaces, surfaces with an elliptic fibration; quasielliptic surfaces constitute a modification this idea that occurs in finite characteristic
Raynaud surfaces and generalized Raynaud surfaces, certain quasielliptic counterexamples to the conclusions of the Kodaira vanishing theorem
Exceptional surfaces, surfaces whose Picard number achieve the bound set by the central Hodge number h1,1
Kähler surfaces, complex surfaces with a Kähler metric; equivalently, surfaces for which the first Betti number b1 is even
Minimal surfaces, surfaces that can't be obtained from another by blowing up at a point; they have no connection with the minimal surfaces of differential geometry
Nodal surfaces, surfaces whose only singularities are nodes
Cayley's nodal cubic, which has 4 nodes
Kummer surfaces, quartic surfaces with 16 nodes
Togliatti surface, a certain quintic with 31 nodes
Barth surfaces, referring to a certain sextic with 65 nodes and decic with 345 nodes
Labs surface, a certain septic with 99 nodes
Endrass surface, a certain surface of degree 8 with 168 nodes
Sarti surface, a certain surface of degree 12 with 600 nodes
Quotient surfaces, surfaces that are constructed as the orbit space of some other surface by the action of a finite group; examples include Kummer, Godeaux, Hopf, and Inoue surfaces
Zariski surfaces, surfaces in finite characteristic that admit a purely inseparable dominant rational map from the projective plane




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.

Kata Kunci Pencarian:

• Geometri kompleks
• Daftar bentuk matematika
• List of complex and algebraic surfaces
• Algebraic surface
• Complex geometry
• K3 surface
• Algebraic geometry and analytic geometry
• List of algebraic geometry topics
• Enriques–Kodaira classification
• Genus (mathematics)
• Kodaira dimension
• List of theorems