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 curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change.
The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context). The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in F-theory.
Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds and smooth 4-manifolds. They are similar to (have analogies with, that is), elliptic curves over number fields.
In mathematics, an ellipticsurface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic...
mathematics, a hyperelliptic surface, or bi-ellipticsurface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the...
plane). A paraboloid is either elliptic or hyperbolic. Equivalently, a paraboloid may be defined as a quadric surface that is not a cylinder, and has...
mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined...
Weierstrass elliptic function. Likewise, genus g surfaces have Riemann surface structures, as (compactifications of) hyperelliptic surfaces y2 = Q(x),...
hyperbola is called an elliptic cylinder, parabolic cylinder and hyperbolic cylinder, respectively. These are degenerate quadric surfaces. When the principal...
In algebraic geometry, an elliptic singularity of a surface, introduced by Wagreich (1970), is a surface singularity such that the arithmetic genus of...
hyperbolic paraboloid shape. Saddle surfaces have negative Gaussian curvature which distinguish them from convex/ellipticalsurfaces which have positive Gaussian...
Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel...
general elliptic K3 surface has exactly 24 singular fibers, each of type I 1 {\displaystyle I_{1}} (a nodal cubic curve). Whether a K3 surface is elliptic can...
genus 1 surfaces A torus of genus 1 An elliptic curve The term double torus is occasionally used to denote a genus 2 surface. A non-orientable surface of genus...
base, it is called a frustum. An elliptical cone is a cone with an elliptical base. A generalized cone is the surface created by the set of lines passing...
1 (an abelian surface) has Kodaira dimension 0; the product of a curve of genus 1 with a curve of genus at least 2 (an ellipticsurface) has Kodaira dimension...
provides a basis (up to torsion) for the Mordell–Weil group of an ellipticsurface E → S {\displaystyle E\to S} , where S {\displaystyle S} is isomorphic...
In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the...
In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas,...
hyperelliptic surfaces κ = 1: ellipticsurfaces κ = 2: surfaces of general type. For more examples see the list of algebraic surfaces. The first five examples...
trivial square. Enriques surfaces are all projective (and therefore Kähler over the complex numbers) and are ellipticsurfaces of genus 0. Over fields...
In electrodynamics, elliptical polarization is the polarization of electromagnetic radiation such that the tip of the electric field vector describes an...
In mathematics, an abelian surface is a 2-dimensional abelian variety. One-dimensional complex tori are just elliptic curves and are all algebraic, but...
obtains an elliptic cone (also called a conical quadric or quadratic cone), which is a special case of a quadric surface. A conical surface S {\displaystyle...
modular surface is one of the ellipticsurfaces studied by Shioda (1972). Barth, Wolf; Hulek, Klaus (1985), "Projective models of Shioda modular surfaces",...
named for Michel Raynaud (1978). To be precise, a Raynaud surface is a quasi-ellipticsurface over an algebraic curve of genus g greater than 1, such that...
In mathematics, Dolgachev surfaces are certain simply connected ellipticsurfaces, introduced by Igor Dolgachev (1981). They can be used to give examples...
Global Information