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Projective Morphism
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Glossary Of Commutative Algebra
This is a glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary of ring theory and glossary of module theory. In this article, all rings are assumed to be commutative with identity 1. !$@ A B C D E F G H . I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmic Kähler Differential
In contexts including complex manifolds and algebraic geometry, a logarithmic differential form is a meromorphic differential form with poles of a certain kind. The concept was introduced by Deligne. Let ''X'' be a complex manifold, ''D'' ⊂ ''X'' a divisor, and ω a holomorphic ''p''-form on ''X''−''D''. If ω and ''d''ω have a pole of order at most one along ''D'', then ω is said to have a logarithmic pole along ''D''. ω is also known as a logarithmic ''p''-form. The logarithmic ''p''-forms make up a subsheaf of the meromorphic ''p''-forms on ''X'' with a pole along ''D'', denoted :\Omega^p_X(\log D). In the theory of Riemann surfaces, one encounters logarithmic one-forms which have the local expression :\omega = \frac =\left(\frac + \frac\right)dz for some meromorphic function (resp. rational function) f(z) = z^mg(z) , where ''g'' is holomorphic and non-vanishing at 0, and ''m'' is the order of ''f'' at ''0''. That is, for some open covering, there are local repr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Variety
In algebraic geometry, an affine variety, or affine algebraic variety, over an algebraically closed field is the zero-locus in the affine space of some finite family of polynomials of variables with coefficients in that generate a prime ideal. If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety. Some texts do not require a prime ideal, and call ''irreducible'' an algebraic variety defined by a prime ideal. This article refers to zero-loci of not necessarily prime ideals as affine algebraic sets. In some contexts, it is useful to distinguish the field in which the coefficients are considered, from the algebraically closed field (containing ) over which the zero-locus is considered (that is, the points of the affine variety are in ). In this case, the variety is said ''defined over'' , and the points of the variety that belong to are said '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead '' displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Divisor
In mathematics, the canonical bundle of a non-singular algebraic variety V of dimension n over a field is the line bundle \,\!\Omega^n = \omega, which is the ''n''th exterior power of the cotangent bundle Ω on ''V''. Over the complex numbers, it is the determinant bundle of holomorphic ''n''-forms on ''V''. This is the dualising object for Serre duality on ''V''. It may equally well be considered as an invertible sheaf. The canonical class is the divisor class of a Cartier divisor ''K'' on ''V'' giving rise to the canonical bundle — it is an equivalence class for linear equivalence on ''V'', and any divisor in it may be called a canonical divisor. An anticanonical divisor is any divisor −''K'' with ''K'' canonical. The anticanonical bundle is the corresponding inverse bundle ω−1. When the anticanonical bundle of V is ample, V is called a Fano variety. The adjunction formula Suppose that ''X'' is a smooth variety and that ''D'' is a smooth diviso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjunction Formula
In mathematics, especially in algebraic geometry and the theory of complex manifolds, the adjunction formula relates the canonical bundle of a variety and a hypersurface inside that variety. It is often used to deduce facts about varieties embedded in well-behaved spaces such as projective space or to prove theorems by induction. Adjunction for smooth varieties Formula for a smooth subvariety Let ''X'' be a smooth algebraic variety or smooth complex manifold and ''Y'' be a smooth subvariety of ''X''. Denote the inclusion map by ''i'' and the ideal sheaf of ''Y'' in ''X'' by \mathcal. The conormal exact sequence for ''i'' is :0 \to \mathcal/\mathcal^2 \to i^*\Omega_X \to \Omega_Y \to 0, where Ω denotes a cotangent bundle. The determinant of this exact sequence is a natural isomorphism :\omega_Y = i^*\omega_X \otimes \operatorname(\mathcal/\mathcal^2)^\vee, where \vee denotes the dual of a line bundle. The particular case of a smooth divisor Suppose that ''D'' is a smooth d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dualizing Sheaf
In algebraic geometry, the dualizing sheaf on a proper scheme ''X'' of dimension ''n'' over a field ''k'' is a coherent sheaf \omega_X together with a linear functional :t_X: \operatorname^n(X, \omega_X) \to k that induces a natural isomorphism of vector spaces :\operatorname_X(F, \omega_X) \simeq \operatorname^n(X, F)^*, \, \varphi \mapsto t_X \circ \varphi for each coherent sheaf ''F'' on ''X'' (the superscript * refers to a dual vector space). The linear functional t_X is called a trace morphism. A pair (\omega_X, t_X), if it is exists, is unique up to a natural isomorphism. In fact, in the language of category theory, \omega_X is an object representing the contravariant functor F \mapsto \operatorname^n(X, F)^* from the category of coherent sheaves on ''X'' to the category of ''k''-vector spaces. For a normal projective variety ''X'', the dualizing sheaf exists and it is in fact the canonical sheaf: \omega_X = \mathcal_X(K_X) where K_X is a canonical divisor. More generally, ... [...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 fr ... [...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] |
Berkovich Spectrum
In mathematics, a Berkovich space, introduced by , is a version of an analytic space over a non-Archimedean field (e.g. ''p''-adic field), refining Tate's notion of a rigid analytic space. Motivation In the complex case, algebraic geometry begins by defining the complex affine space to be \Complex^n. For each U\subset\Complex^n, we define \mathcal_U, the ring of analytic functions on U to be the ring of holomorphic functions, i.e. functions on U that can be written as a convergent power series in a neighborhood of each point. We then define a local model space for f_, \ldots, f_\in\mathcal_U to be :X:=\ with \mathcal_X=\mathcal_U/(f_, \ldots,f_). A complex analytic space is a locally ringed \Complex-space (Y, \mathcal_Y) which is locally isomorphic to a local model space. When k is a complete non-Archimedean field, we have that k is totally disconnected. In such a case, if we continue with the same definition as in the complex case, we wouldn't get a good analytic theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Spec
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal. Zariski topology For any ideal ''I'' of ''R'', define V_I to be the set of prime ideals containing ''I''. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''''f'' is an open subset of \operatorname(R), and \ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Spectrum
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal. Zariski topology For any ideal ''I'' of ''R'', define V_I to be the set of prime ideals containing ''I''. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''''f'' is an open subset of \operatorname(R), and \ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
