Category: Algebraic surfaces

Castelnuovo–de Franchis theorem
In mathematics, the Castelnuovo–de Franchis theorem is a classical result on complex algebraic surfaces. Let X be such a surface, projective and non-singular, and let ω1 and ω2 be two differentials of
Dolgachev surface
In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by Igor Dolgachev. They can be used to give examples of an infinite family of homeomorphic simply connecte
Hyperelliptic surface
In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elli
Supersingular K3 surface
In algebraic geometry, a supersingular K3 surface is a K3 surface over a field k of characteristic p > 0 such that the slopes of Frobenius on the crystalline cohomology H2(X,W(k)) are all equal to 1.
Castelnuovo surface
In mathematics, a Castelnuovo surface is a surface of general type such that the canonical bundle is very ample and such that c12 = 3pg − 7. Guido Castelnuovo proved that if the canonical bundle is ve
Cayley's nodal cubic surface
In algebraic geometry, the Cayley surface, named after Arthur Cayley, is a cubic nodal surface in 3-dimensional projective space with four conical points. It can be given by the equation when the four
Wave surface
In mathematics, Fresnel's wave surface, found by Augustin-Jean Fresnel in 1822, is a quartic surface describing the propagation of light in an optically biaxial crystal. Wave surfaces are special case
Veronese surface
In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complet
Elliptic surface
In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth cur
Quaternary cubic
In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space.
Conical surface
In geometry, a (general) conical surface is the unbounded surface formed by the union of all the straight lines that pass through a fixed point — the apex or vertex — and any point of some fixed space
Nodal surface
In algebraic geometry, a nodal surface is a surface in (usually complex) projective space whose only singularities are nodes. A major problem about them is to find the maximum number of nodes of a nod
Quartic surface
In mathematics, especially in algebraic geometry, a quartic surface is a surface defined by an equation of degree 4. More specifically there are two closely related types of quartic surface: affine an
Riemann–Roch theorem for surfaces
In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by Castelnuovo , after preliminary ver
Tetrahedroid
In algebraic geometry, a tetrahedroid (or tétraédroïde) is a special kind of Kummer surface studied by Cayley, with the property that the intersections with the faces of a fixed tetrahedron are given
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a grou
Enneper surface
In differential geometry and algebraic geometry, the Enneper surface is a self-intersecting surface that can be described parametrically by: It was introduced by Alfred Enneper in 1864 in connection w
Sarti surface
In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by Alessandra Sarti. The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), th
Zariski surface
In algebraic geometry, a branch of mathematics, a Zariski surface is a surface over a field of characteristic p > 0 such that there is a dominant inseparable map of degree p from the projective plane
Labs septic
In mathematics, the Labs septic surface is a degree-7 (septic) nodal surface with 99 nodes found by . As of 2015, it has the largest known number of nodes of a degree-7 surface, though this number is
Godeaux surface
In mathematics, a Godeaux surface is one of the surfaces of general type introduced by Lucien Godeaux in 1931.Other surfaces constructed in a similar way with the same Hodge numbers are also sometimes
Bordiga surface
In algebraic geometry, a Bordiga surface is a certain sort of rational surface of degree 6 in P4, introduced by Giovanni Bordiga. A Bordiga surface is isomorphic to the projective plane blown up in 10
Cox–Zucker machine
The Cox–Zucker machine is an algorithm created by David A. Cox and Steven Zucker. This algorithm determines whether a given set of sections provides a basis (up to torsion) for the Mordell–Weil group
Algebraic surface
In mathematics, an algebraic surface is an algebraic variety of dimension two. In the case of geometry over the field of complex numbers, an algebraic surface has complex dimension two (as a complex m
Shioda modular surface
In mathematics, a Shioda modular surface is one of the elliptic surfaces studied by Shioda.
Châtelet surface
In algebraic geometry, a Châtelet surface is a rational surface studied by Châtelet given by an equation where P has degree 3 or 4. They are conic bundles.
Horikawa surface
In mathematics, a Horikawa surface is one of the surfaces of general type introduced by Horikawa.These are surfaces with q = 0 and pg = c12/2 + 2 or c12/2 + 3/2 (which implies that they are more or le
Abelian surface
In mathematics, an abelian surface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but Riemann discovered that most complex tori of dim
White surface
In algebraic geometry, a White surface is one of the rational surfaces in Pn studied by , generalizing cubic surfaces and Bordiga surfaces, which are the cases n = 3 or 4. A White surface in Pn is giv
Clebsch surface
In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional linescan be defined over the rea
Barth surface
In algebraic geometry, a Barth surface is one of the complex nodal surfaces in 3 dimensions with large numbers of double points found by Wolf Barth. Two examples are the Barth sextic of degree 6 with
Burniat surface
In mathematics, a Burniat surface is one of the surfaces of general type introduced by Pol Burniat.
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.
Pinch point (mathematics)
In geometry, a pinch point or cuspidal point is a type of singular point on an algebraic surface. The equation for the surface near a pinch point may be put in the form where [4] denotes terms of degr
Nagata–Biran conjecture
In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces.
Umbral moonshine
In mathematics, umbral moonshine is a mysterious connection between Niemeier lattices and Ramanujan's mock theta functions. It is a generalization of the Mathieu moonshine phenomenon connecting repres
Segre surface
In algebraic geometry, a Segre surface, studied by Corrado Segre and Beniamino Segre, is an intersection of two quadrics in 4-dimensional projective space.They are rational surfaces isomorphic to a pr
Enriques–Kodaira classification
In mathematics, the Enriques–Kodaira classification is a classification of compact complex surfaces into ten classes. For each of these classes, the surfaces in the class can be parametrized by a modu
Del Pezzo surface
In mathematics, a del Pezzo surface or Fano surface is a two-dimensional Fano variety, in other words a non-singular projective algebraic surface with ample anticanonical divisor class. They are in so
Zeuthen–Segre invariant
In algebraic geometry, the Zeuthen–Segre invariant I is an invariant of a projective surface found in a complex projective space which was introduced by Zeuthen and rediscovered by Corrado Segre. The
Fake projective plane
In mathematics, a fake projective plane (or Mumford surface) is one of the 50 complex algebraic surfaces that have the same Betti numbers as the projective plane, but are not isomorphic to it. Such ob
Riemann–Roch theorem for smooth manifolds
In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making t
Hilbert modular variety
In mathematics, a Hilbert modular surface or Hilbert–Blumenthal surface is an algebraic surface obtained by taking a quotient of a product of two copies of the upper half-plane by a Hilbert modular gr
Brill–Noether theory
In algebraic geometry, Brill–Noether theory, introduced by Alexander von Brill and Max Noether, is the study of special divisors, certain divisors on a curve C that determine more compatible functions
Noether inequality
In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It hold
Beauville surface
In mathematics, a Beauville surface is one of the surfaces of general type introduced by Arnaud Beauville . They are examples of "fake quadrics", with the same Betti numbers as quadric surfaces.
Hirzebruch surface
In mathematics, a Hirzebruch surface is a ruled surface over the projective line. They were studied by Friedrich Hirzebruch.
Weddle surface
In algebraic geometry, a Weddle surface, introduced by Thomas Weddle , is a quartic surface in 3-dimensional projective space, given by the locus of vertices of the family of cones passing through 6 p
Kummer surface
In algebraic geometry, a Kummer quartic surface, first studied by Ernst Kummer, is an irreducible nodal surface of degree 4 in with the maximal possible number of 16 double points. Any such surface is
Picard modular surface
In mathematics, a Picard modular surface, studied by Picard, is a complex surface constructed as a quotient of the unit ball in C2 by a Picard modular group.Picard modular surfaces are some of the sim
Reider's theorem
In algebraic geometry, Reider's theorem gives conditions for a line bundle on a projective surface to be very ample.
Hodge index theorem
In mathematics, the Hodge index theorem for an algebraic surface V determines the signature of the intersection pairing on the algebraic curves C on V. It says, roughly speaking, that the space spanne
Thom conjecture
In mathematics, a smooth algebraic curve in the complex projective plane, of degree , has genus given by the genus–degree formula . The Thom conjecture, named after French mathematician René Thom, sta
Plücker surface
In algebraic geometry, a Plücker surface, studied by Julius Plücker, is a quartic surface in 3-dimensional projective space with a double line and 8 nodes.
Catanese surface
In mathematics, a Catanese surface is one of the surfaces of general type introduced by.
Todorov surface
In algebraic geometry, a Todorov surface is one of a class of surfaces of general type introduced by Todorov for which the conclusion of the Torelli theorem does not hold.
Elliptic singularity
In algebraic geometry, an elliptic singularity of a surface, introduced by , is a surface singularity such that the arithmetic genus of its local ring is 1.
Togliatti surface
In algebraic geometry, a Togliatti surface is a nodal surface of degree five with 31 nodes. The first examples were constructed by Eugenio G. Togliatti. Arnaud Beauville proved that 31 is the maximum
Complex projective plane
In mathematics, the complex projective plane, usually denoted P2(C), is the two-dimensional complex projective space. It is a complex manifold of complex dimension 2, described by three complex coordi
Irregularity of a surface
In mathematics, the irregularity of a complex surface X is the Hodge number , usually denoted by q. The irregularity of an algebraic surface is sometimes defined to be this Hodge number, and sometimes
Coble surface
In algebraic geometry, a Coble surface was defined by to be a smooth rational projective surface with empty anti-canonical linear system |−K| and non-empty anti-bicanonical linear system |−2K|. An exa
Castelnuovo's contraction theorem
In mathematics, Castelnuovo's contraction theorem is used in the classification theory of algebraic surfaces to construct the minimal model of a given smooth algebraic surface. More precisely, let be
Cubic surface
In mathematics, a cubic surface is a surface in 3-dimensional space defined by one polynomial equation of degree 3. Cubic surfaces are fundamental examples in algebraic geometry. The theory is simplif
Endrass surface
In algebraic geometry, an Endrass surface is a nodal surface of degree 8 with 168 real nodes, found by Stephan Endrass. As of 2007, it remained the record-holder for the most number of real nodes for
Cayley's ruled cubic surface
In differential geometry, Cayley's ruled cubic surface is the ruled cubic surface It contains a nodal line of self-intersection and two cuspital points at infinity. In projective coordinates it is .
Rational surface
In algebraic geometry, a branch of mathematics, a rational surface is a surface birationally equivalent to the projective plane, or in other words a rational variety of dimension two. Rational surface
Bogomolov–Miyaoka–Yau inequality
In mathematics, the Bogomolov–Miyaoka–Yau inequality is the inequality between Chern numbers of compact complex surfaces of general type. Its major interest is the way it restricts the possible topolo
Raynaud surface
In mathematics, a Raynaud surface is a particular kind of algebraic surface that was introduced in William E. Lang and named for Michel Raynaud. To be precise, a Raynaud surface is a quasi-elliptic su
Fermat cubic
In geometry, the Fermat cubic, named after Pierre de Fermat, is a surface defined by Methods of algebraic geometry provide the following parameterization of Fermat's cubic: In projective space the Fer
Du Val singularity
In algebraic geometry, a Du Val singularity, also called simple surface singularity, Kleinian singularity, or rational double point, is an isolated singularity of a complex surface which is modeled on
Fano surface
In algebraic geometry, a Fano surface is a surface of general type (in particular, not a Fano variety) whose points index the lines on a non-singular cubic threefold. They were first studied by Fano.
Rational singularity
In mathematics, more particularly in the field of algebraic geometry, a scheme has rational singularities, if it is normal, of finite type over a field of characteristic zero, and there exists a prope
Noether's theorem on rationality for surfaces
In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surfa
K3 surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means
Peano surface
In mathematics, the Peano surface is the graph of the two-variable function It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and min
Ruled surface
In geometry, a surface S is ruled (also called a scroll) if through every point of S there is a straight line that lies on S. Examples include the plane, the lateral surface of a cylinder or cone, a c
Humbert surface
In algebraic geometry, a Humbert surface, studied by Humbert, is a surface in the moduli space of principally polarized abelian surfaces consisting of the surfaces with a symmetric endomorphism of som
Surface of general type
In algebraic geometry, a surface of general type is an algebraic surface with Kodaira dimension 2. Because of Chow's theorem any compact complex manifold of dimension 2 and with Kodaira dimension 2 wi
Barlow surface
In mathematics, a Barlow surface is one of the complex surfaces introduced by . They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a project
Campedelli surface
In mathematics, a Campedelli surface is one of the surfaces of general type introduced by Campedelli.Surfaces with the same Hodge numbers are called numerical Campedelli surfaces.
Enriques surface
In mathematics, Enriques surfaces are algebraic surfaces such that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square. Enriques surfaces are all projective (a