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Néron Model
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety ''AK'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''AK'' from Spec(''K'') to Spec(''R''), in other words the "best possible" group scheme ''AR'' defined over ''R'' corresponding to ''AK''. They were introduced by for abelian varieties over the quotient field of a Dedekind domain ''R'' with perfect residue fields, and extended this construction to semiabelian varieties over all Dedekind domains. Definition Suppose that ''R'' is a Dedekind domain with field of fractions ''K'', and suppose that ''AK'' is a smooth separated scheme over ''K'' (such as an abelian variety). Then a Néron model of ''AK'' is defined to be a smooth separated scheme ''AR'' over ''R'' with fiber ''AK'' that is universal in the following sense. :If ''X'' is a smooth separated scheme over ''R'' then any ''K''-morphism from ''X''''K'' to ''AK'' ca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros. The fundamental objects of study in algebraic geometry are algebraic varieties, which are geometric manifestations of solutions of systems of polynomial equations. Examples of the most studied classes of algebraic varieties are: plane algebraic curves, which include lines, circles, parabolas, ellipses, hyperbolas, cubic curves like elliptic curves, and quartic curves like lemniscates and Cassini ovals. A point of the plane belongs to an algebraic curve if its coordinates satisfy a given polynomial equation. Basic questions involve the study of the points of special interest like the singular points, the inflection points and the points at infinity. More advanced questions involve the topology of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Morphism
In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if *(i) it is locally of finite presentation *(ii) it is flat, and *(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular. (iii) means that each geometric fiber of ''f'' is a nonsingular variety (if it is separated). Thus, intuitively speaking, a smooth morphism gives a flat family of nonsingular varieties. If ''S'' is the spectrum of an algebraically closed field and ''f'' is of finite type, then one recovers the definition of a nonsingular variety. Equivalent definitions There are many equivalent definitions of a smooth morphism. Let f: X \to S be locally of finite presentation. Then the following are equivalent. # ''f'' is smooth. # ''f'' is formally smooth (see below). # ''f'' is flat and the sheaf of relative differentials \Omega_ is locally free of rank equal to the relative dimension of X/S. # For any x \in X, there exists a neighborhood \operatornameB of x and a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Separated Morphism
In algebraic geometry, given a morphism of schemes p: X \to S, the diagonal morphism :\delta: X \to X \times_S X is a morphism determined by the universal property of the fiber product X \times_S X of ''p'' and ''p'' applied to the identity 1_X : X \to X and the identity 1_X. It is a special case of a graph morphism: given a morphism f: X \to Y over ''S'', the graph morphism of it is X \to X \times_S Y induced by f and the identity 1_X. The diagonal embedding is the graph morphism of 1_X. By definition, ''X'' is a separated scheme over ''S'' (p: X \to S is a separated morphism) if the diagonal morphism is a closed immersion. Also, a morphism p: X \to S locally of finite presentation is an unramified morphism if and only if the diagonal embedding is an open immersion. Explanation As an example, consider an algebraic variety over an algebraically closed field ''k'' and p: X \to \operatorname(k) the structure map. Then, identifying ''X'' with the set of its ''k''-rational points, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The Category (mathematics), category of group schemes is somewhat better behaved than that of Group variety, group varieties, since all homomorphisms have Kernel (category theory), kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The ini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Point
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Group
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Scheme
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 group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally from abe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate's Algorithm
In the theory of elliptic curves, Tate's algorithm takes as input an integral model of an elliptic curve ''E'' over \mathbb, or more generally an algebraic number field, and a prime or prime ideal ''p''. It returns the exponent ''f''''p'' of ''p'' in the conductor of ''E'', the type of reduction at ''p'', the local index : c_p= (\mathbb_p):E^0(\mathbb_p) where E^0(\mathbb_p) is the group of \mathbb_p-points whose reduction mod ''p'' is a non-singular point. Also, the algorithm determines whether or not the given integral model is minimal at ''p'', and, if not, returns an integral model with integral coefficients for which the valuation at ''p'' of the discriminant is minimal. Tate's algorithm also gives the structure of the singular fibers given by the Kodaira symbol or Néron symbol, for which, see elliptic surfaces: in turn this determines the exponent ''f''''p'' of the conductor ''E''. Tate's algorithm can be greatly simplified if the characteristic of the residue class fi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 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-manifo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minimal Model Program
In algebraic geometry, the minimal model program is part of the birational classification of algebraic varieties. Its goal is to construct a birational model of any complex projective variety which is as simple as possible. The subject has its origins in the classical birational geometry of surfaces studied by the Italian school, and is currently an active research area within algebraic geometry. Outline The basic idea of the theory is to simplify the birational classification of varieties by finding, in each birational equivalence class, a variety which is "as simple as possible". The precise meaning of this phrase has evolved with the development of the subject; originally for surfaces, it meant finding a smooth variety X for which any birational morphism f\colon X \to X' with a smooth surface X' is an isomorphism. In the modern formulation, the goal of the theory is as follows. Suppose we are given a projective variety X, which for simplicity is assumed non-singular. There are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
