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Equations Defining Abelian Varieties
In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The equations defining abelian varieties are a topic of study because every abelian variety is a projective variety. In dimension ''d'' ≥ 2, however, it is no longer as straightforward to discuss such equations. There is a large classical literature on this question, which in a reformulation is, for complex algebraic geometry, a question of describing relations between theta functions. The modern geometric treatment now refers to some basic papers of David Mumford, from 1966 to 1967, which reformulated that theory in terms from abstract algebraic geometry valid over general fields. Complete intersections The only 'easy' cases are those for ''d'' = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In ''P''3, an elliptic curve can be obtained as the intersection of two quad ... [...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] |
Very Ample
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety ''X'' amounts to understanding the different ways of mapping ''X'' into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor. In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polynomial Algebra
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field. Often, the term "polynomial ring" refers implicitly to the special case of a polynomial ring in one indeterminate over a field. The importance of such polynomial rings relies on the high number of properties that they have in common with the ring of the integers. Polynomial rings occur and are often fundamental in many parts of mathematics such as number theory, commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings, rings of formal power series, Ore polynomials, graded rings, have been introduced for generalizing some properties of polynomial rings. A closely related notion is that of the ring ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quotient Ring
In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring and a two-sided ideal in , a new ring, the quotient ring , is constructed, whose elements are the cosets of in subject to special and operations. (Only the fraction slash "/" is used in quotient ring notation, not a horizontal fraction bar.) Quotient rings are distinct from the so-called "quotient field", or field of fractions, of an integral domain as well as from the more general "rings of quotients" obtained by localization. Formal quotient ring construction Given a ring and a two-sided ideal in , we may define an equivalence relation on as follows: : if and only if is in . Using the ideal properties, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted v \otimes w. An element of the form v \otimes w is called the tensor product of and . An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for and , a basis of V \otimes W is formed by all tensor products of a basis element of and a basis element of . The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space factors uniquely through a linear map V\otimes W\to Z (see Universal property). Tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Commutative Ring
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition). It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded \Z-algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra. First properties Generally, the index set of a graded ring is assumed to be the set of nonnegative integers, unless otherwise explicitly specified. This is the case in this articl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinate Ring
In algebraic geometry, the homogeneous coordinate ring ''R'' of an algebraic variety ''V'' given as a subvariety of projective space of a given dimension ''N'' is by definition the quotient ring :''R'' = ''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N''thinsp;/''I'' where ''I'' is the homogeneous ideal defining ''V'', ''K'' is the algebraically closed field over which ''V'' is defined, and :''K'' 'X''0, ''X''1, ''X''2, ..., ''X''''N'' is the polynomial ring in ''N'' + 1 variables ''X''''i''. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra. Formulation Since ''V'' is assumed to be a variety, and so an irreducible algebraic set, the ideal ''I'' can be chosen to be a prime ideal, and so ''R'' is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Characteristic
In mathematics, a theta characteristic of a non-singular algebraic curve ''C'' is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles ''L'' on a connected compact Riemann surface, it is therefore ''L'' such that ''L''2 is the canonical bundle, here also equivalently the holomorphic cotangent bundle. In terms of algebraic geometry, the equivalent definition is as an invertible sheaf, which squares to the sheaf of differentials of the first kind. Theta characteristics were introduced by History and genus 1 The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stone–von Neumann Theorem
In mathematics and in theoretical physics, the Stone–von Neumann theorem refers to any one of a number of different formulations of the uniqueness of the canonical commutation relations between position and momentum operators. It is named after Marshall Stone and John von Neumann. Representation issues of the commutation relations In quantum mechanics, physical observables are represented mathematically by linear operators on Hilbert spaces. For a single particle moving on the real line \mathbb, there are two important observables: position and momentum. In the Schrödinger representation quantum description of such a particle, the position operator and momentum operator p are respectively given by \begin[] [x \psi](x_0) &= x_0 \psi(x_0) \\[] [p \psi](x_0) &= - i \hbar \frac(x_0) \end on the domain V of infinitely differentiable functions of compact support on \mathbb. Assume \hbar to be a fixed ''non-zero'' real number—in quantum theory \hbar is the reduced Planck' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Character
A protagonist () is the main character of a story. The protagonist makes key decisions that affect the plot, primarily influencing the story and propelling it forward, and is often the character who faces the most significant obstacles. If a story contains a subplot, or is a narrative made up of several stories, then each subplot may have its own protagonist. The protagonist is the character whose fate is most closely followed by the reader or audience, and who is opposed by the antagonist. The antagonist provides obstacles and complications and creates conflicts that test the protagonist, revealing the strengths and weaknesses of the protagonist's character, and having the protagonist develop as a result. Etymology The term ''protagonist'' comes , combined of (, 'first') and (, 'actor, competitor'), which stems from (, 'contest') via (, 'I contend for a prize'). Ancient Greece The earliest known examples of a protagonist are found in Ancient Greece. At first, dramatic pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil Pairing
Weil may refer to: Places in Germany *Weil, Bavaria *Weil am Rhein, Baden-Württemberg *Weil der Stadt, Baden-Württemberg *Weil im Schönbuch, Baden-Württemberg Other uses * Weil (river), Hesse, Germany * Weil (surname), including people with the surname Weill, Weyl * Doctor Weil (Mega Man Zero), a fictional character from the ''Mega Man'' Zero video game series * Weil-Marbach, now the Marbach Stud in Baden-Württemberg See also * Weill (other) * Weil, Gotshal & Manges Weil, Gotshal & Manges LLP is an American international law firm with approximately 1,100 attorneys, headquartered in New York City. With a gross annual revenue in excess of $1.8 billion, it is among the world's largest law firms according to ..., law firm founded in the United States * Weil's disease {{disambiguation, geo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
