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Analytic Varieties
In mathematics, and in particular differential geometry and complex geometry, a complex analytic variety Complex analytic variety (or just variety) is sometimes required to be irreducible and (or) reduced or complex analytic space is a generalization of a complex manifold which allows the presence of singularities. Complex analytic varieties are locally ringed spaces which are locally isomorphic to local model spaces, where a local model space is an open subset of the vanishing locus of a finite set of holomorphic functions. Definition Denote the constant sheaf on a topological space with value \mathbb by \underline. A \mathbb-space is a locally ringed space (X, \mathcal_X), whose structure sheaf is an algebra over \underline. Choose an open subset U of some complex affine space \mathbb^n, and fix finitely many holomorphic functions f_1,\dots,f_k in U. Let X=V(f_1,\dots,f_k) be the common vanishing locus of these holomorphic functions, that is, X=\. Define a sheaf of rings o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
BY-NC-SA
A Creative Commons (CC) license is one of several public copyright licenses that enable the free distribution of an otherwise copyrighted "work".A "work" is any creative material made by a person. A painting, a graphic, a book, a song/lyrics to a song, or a photograph of almost anything are all examples of "works". A CC license is used when an author wants to give other people the right to share, use, and build upon a work that the author has created. CC provides an author flexibility (for example, they might choose to allow only non-commercial uses of a given work) and protects the people who use or redistribute an author's work from concerns of copyright infringement as long as they abide by the conditions that are specified in the license by which the author distributes the work. There are several types of Creative Commons licenses. Each license differs by several combinations that condition the terms of distribution. They were initially released on December 16, 2002, by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales De L'Institut Fourier
The ''Annales de l'Institut Fourier'' is a French mathematical journal publishing papers in all fields of mathematics. It was established in 1949. The journal publishes one volume per year, consisting of six issues. The current editor-in-chief is Hervé Pajot. Articles are published either in English or in French. The journal is indexed in ''Mathematical Reviews'', ''Zentralblatt MATH'' and the Web of Science. According to the ''Journal Citation Reports'', the journal had a 2008 impact factor of 0.804. 2008 Journal Citation Reports, Science Edition, Thomson Scientific Thomson Scientific was one of the six (later five) strategic business units of The Thomson Corporation, beginning in 2007, after being separated from Thomson Scientific & Healthcare. Following the merger of Thomson with Reuters Group to form Thom ..., 2008. References External links * Mathematics journals Academic journals established in 1949 Multilingual journals Bimonthly journals Open access journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GAGA
Gaga ( he, גע גע literally 'touch touch') (also: ga-ga, gaga ball, or ga-ga ball) is a variant of dodgeball that is played in a gaga "pit". The game combines dodging, striking, running, and jumping, with the objective of being the last person standing. Players hit the ball at each other with their hands, and are eliminated if the ball strikes them on or below the knee. The game can be played by a group of individual players or with teams, as well as in one-on-one matches. Rules, ball types, pit surfaces, and pit sizes can vary widely at different venues. Gameplay Gaga is played in a large fenced-in area (usually an octagon or hexagon) called a gaga pit. The gaga pit generally consists of flat walls atop a smooth dirt, turf, sand, or rubberized surface. The gaga ball can vary in size and form, generally ranging from a foam dodgeball to a rubber kickball. In some games, the ball may not actually take the shape of a sphere if the proper equipment is not available. The game begin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Algebraic Variety
In algebraic geometry, a complex algebraic variety is an algebraic variety (in the scheme sense or otherwise) over the field of complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...s.Aleksei Nikolaevich Parshin, Parshin, Alexei N., and Igor Rostislavovich Shafarevich, eds. ''Algebraic Geometry III: Complex Algebraic Varieties. Algebraic Curves and Their Jacobians.'' Vol. 3. Springer, 1998. Chow's theorem Algebraic geometry and analytic geometry#Chow.27s theorem, Chow's theorem states that a projective analytic variety; i.e., a closed analytic subvariety of the complex projective space \mathbb\mathbf^n is an algebraic variety; it is usually simply referred to as a projective variety. Relation with similar concepts Not every complex analytic variety is algebraic, though ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Analytic Space
An analytic space is a generalization of an analytic manifold that allows singularities. An analytic space is a space that is locally the same as an analytic variety. They are prominent in the study of several complex variables, but they also appear in other contexts. Definition Fix a field ''k'' with a valuation. Assume that the field is complete and not discrete with respect to this valuation. For example, this includes R and C with respect to their usual absolute values, as well as fields of Puiseux series with respect to their natural valuations. Let ''U'' be an open subset of ''k''''n'', and let ''f''1, ..., ''f''''k'' be a collection of analytic functions on ''U''. Denote by ''Z'' the common vanishing locus of ''f''1, ..., ''f''''k'', that is, let ''Z'' = . ''Z'' is an analytic variety. Suppose that the structure sheaf of ''U'' is \mathcal_U. Then ''Z'' has a structure sheaf \mathcal_Z = \mathcal_U / \mathcal_Z, where \mathcal_Z is the ideal generated by ''f''1, . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility. The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gluing Schemes
In algebraic geometry, a new scheme (e.g. an algebraic variety) can be obtained by gluing existing schemes through gluing maps. Statement Suppose there is a (possibly infinite) family of schemes \_ and for pairs i, j, there are open subsets U_ and isomorphisms \varphi_ : U_ \overset\to U_. Now, if the isomorphisms are compatible in the sense: for each i, j, k, # \varphi_ = \varphi_^, # \varphi_(U_ \cap U_) = U_ \cap U_, # \varphi_ \circ \varphi_ = \varphi_ on U_ \cap U_, then there exists a scheme ''X'', together with the morphisms \psi_i : X_i \to X such that # \psi_i is an isomorphism onto an open subset of ''X'', # X = \cup_i \psi_i(X_i), # \psi_i(U_) = \psi_i(X_i) \cap \psi_j(X_j), # \psi_i = \psi_j \circ \varphi_ on U_. Examples Projective line Let X = \operatorname(k \simeq \mathbb^1, Y = \operatorname(k \simeq \mathbb^1 be two copies of the affine line over a field ''k''. Let X_t = \ = \operatorname(k , t^ be the complement of the origin and Y_u = \ defined similar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme Of Finite Type
For a homomorphism ''A'' → ''B'' of commutative rings, ''B'' is called an ''A''-algebra of finite type if ''B'' is a finitely generated as an ''A''-algebra. It is much stronger for ''B'' to be a finite ''A''-algebra, which means that ''B'' is finitely generated as an ''A''-module. For example, for any commutative ring ''A'' and natural number ''n'', the polynomial ring ''A'' 'x''1, ..., ''xn''is an ''A''-algebra of finite type, but it is not a finite ''A''-module unless ''A'' = 0 or ''n'' = 0. Another example of a finite-type morphism which is not finite is \mathbb \to \mathbb x,y]/(y^2 - x^3 - t). The analogous notion in terms of schemes is: a morphism ''f'': ''X'' → ''Y'' of schemes is of finite type if ''Y'' has a covering by affine open subschemes ''Vi'' = Spec ''Ai'' such that ''f''−1(''Vi'') has a finite covering by affine open subschemes ''Uij'' = Spec ''Bij'' with ''Bij'' an ''Ai''-algebra of finite type. One also says that ''X'' is of finite type over ''Y''. For exa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Affine Space
Affine geometry, broadly speaking, is the study of the geometrical properties of lines, planes, and their higher dimensional analogs, in which a notion of "parallel" is retained, but no metrical notions of distance or angle are. Affine spaces differ from linear spaces (that is, vector spaces) in that they do not have a distinguished choice of origin. So, in the words of Marcel Berger, "An affine space is nothing more than a vector space whose origin we try to forget about, by adding translations to the linear maps."* Accordingly, a complex affine space, that is an affine space over the complex numbers, is like a complex vector space, but without a distinguished point to serve as the origin. Affine geometry is one of the two main branches of classical algebraic geometry, the other being projective geometry. A complex affine space can be obtained from a complex projective space by fixing a hyperplane, which can be thought of as a hyperplane of ideal points "at infinity" of the aff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
