|
Theta Divisors
In mathematics, the theta divisor Θ is the divisor in the sense of algebraic geometry defined on an abelian variety ''A'' over the complex numbers (and principally polarized) by the zero locus of the associated Riemann theta-function. It is therefore an algebraic subvariety of ''A'' of dimension dim ''A'' − 1. Classical theory Classical results of Bernhard Riemann describe Θ in another way, in the case that ''A'' is the Jacobian variety ''J'' of an algebraic curve (compact Riemann surface) ''C''. There is, for a choice of base point ''P'' on ''C'', a standard mapping of ''C'' to ''J'', by means of the interpretation of ''J'' as the linear equivalence classes of divisors on ''C'' of degree 0. That is, ''Q'' on ''C'' maps to the class of ''Q'' − ''P''. Then since ''J'' is an algebraic group, ''C'' may be added to itself ''k'' times on ''J'', giving rise to subvarieties ''W''''k''. If ''g'' is the genus of ''C'', Riemann proved that Θ is a translate on ''J'' of ''W'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Genus (mathematics)
In mathematics, genus (: genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic \chi, via the relationship \chi=2-2g for closed surfaces, where g is the genus. For surfaces with b boundary components, the equation reads \chi=2-2g-b. In layman's terms, the genus is the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such hole, while a sphere has 0. The green surface pictured above has 2 holes of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Theta Functions
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. Theta functions are parametrized by points in a tube domain inside a complex Lagrangian Grassmannian, namely the Siegel upper half space. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Exact Sequence
In mathematics, an exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context of group theory, a sequence :G_0\;\xrightarrow\; G_1 \;\xrightarrow\; G_2 \;\xrightarrow\; \cdots \;\xrightarrow\; G_n of groups and group homomorphisms is said to be exact at G_i if \operatorname(f_i)=\ker(f_). The sequence is called exact if it is exact at each G_i for all 1\leq i |
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded the National Medal of Science. He is currently a University Professor Emeritus in the Division of Applied Mathematics at Brown University. Early life and education Mumford was born in Worth, West Sussex in England, of an English father and American mother. His father William started an experimental school in Tanzania and worked for the then newly created United Nations. He attended Phillips Exeter Academy, where he received a Westinghouse Science Talent Search prize for his relay-based computer project. Mumford then went to Harvard University, where he became a student of Oscar Zariski. At Harvard, he became a Putnam Fellow in 1955 and 1956. He completed his PhD in 1961, with a thesis entitled ''Existence of the moduli scheme for curve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
George Kempf
George Rushing Kempf (Globe, Arizona, August 12, 1944 – Lawrence, Kansas, July 16, 2002) was a mathematician who worked on algebraic geometry, who proved the Riemann–Kempf singularity theorem, the Kempf–Ness theorem, the Kempf vanishing theorem, and who introduced Kempf varieties. Mumford on Kempf 'I met George in 1970 when he burst on the algebraic geometry scene with a spectacular PhD thesis. His thesis gave a wonderful analysis of the singularities of the subvarieties W_r of the Jacobian of a curve obtained by adding the curve to itself r times inside its Jacobian. This was one of the major themes that he pursued throughout his career: understanding the interaction of a curve with its Jacobian and especially to the map from the r-fold symmetric product of the curve to the Jacobian. In his thesis he gave a determinantal representation both of W_r and of its tangent cone at all its singular points, which gives you a complete understanding of the nature of these singularit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cartier Divisor
In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields. Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-''r'' subvariety need not be definable by only ''r'' equations when ''r'' is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Holomorphic Line Bundle
In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle. By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety ''X'' (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on ''X''. Definition through trivialization Specifically, one requires that the trivialization maps :\phi_U : \pi^(U) \to U \times \mathbf^k are biholomorphic maps. This is equivalent to requiring that the transition functions :t_ : U\cap V \to \mathrm_k(\mathbf) are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guarant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Global Section
In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as Set (mathematics), sets, abelian groups, Ring (mathematics), rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous function, continuous function (mathematics), functions defined on that open set. Such data are well-behaved in that they can be restricted to smaller open sets, and also the data assigned to an open set are equivalent to all collections of compatible data assigned to collections of smaller open sets cover (topology), covering the original open set (intuitively, every datum is the sum of its constituent data). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract Mathematical object, objects. Their precise definition is rather technical. They are specifically defined as sheaves of sets or as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Multiplicity Of A Point
Multiplicity may refer to: Arts and entertainment * ''Multiplicity'' (film), a 1996 comedy film starring Michael Keaton * ''Multiplicity'' (album), 2005 studio album by Dave Weckl Science * Multiplicity (chemistry), multiplicity in quantum chemistry is a function of angular spin momentum * Multiplicity (informatics), a type of relationship in class diagrams for Unified Modeling Language used in software engineering * Multiplicity (mathematics), the number of times an element is repeated in a multiset * Multiplicity (software), a software application which allows a user to control two or more computers from one mouse and keyboard * Multiplicity (statistical mechanics), the number of microstates corresponding to a particular macrostate in a thermodynamic system, symbolized by the Greek letter Ω * Dissociative identity disorder, psychological condition formerly called "multiple personality disorder" where a person exhibits multiple, distinct overlapping identities * Statistical mu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Linear System Of Divisors
In algebraic geometry, a linear system of divisors is an algebraic generalization of the geometric notion of a family of curves; the dimension of the linear system corresponds to the number of parameters of the family. These arose first in the form of a ''linear system'' of algebraic curves in the projective plane. It assumed a more general form, through gradual generalisation, so that one could speak of linear equivalence of divisor (algebraic geometry), divisors ''D'' on a general Scheme (mathematics), scheme or even a ringed space (X, \mathcal_X). Linear systems of dimension 1, 2, or 3 are called a Pencil (mathematics), pencil, a net, or a web, respectively. A map determined by a linear system is sometimes called the Kodaira map. Definitions Given a general variety X, two divisors D,E \in \text(X) are linearly equivalent if :E = D + (f)\ for some non-zero rational function f on X, or in other words a non-zero element f of the Function field of an algebraic variety, func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Algebraic Curve
In mathematics, an affine algebraic plane curve is the zero set of a polynomial in two variables. A projective algebraic plane curve is the zero set in a projective plane of a homogeneous polynomial in three variables. An affine algebraic plane curve can be completed in a projective algebraic plane curve by homogenization of a polynomial, homogenizing its defining polynomial. Conversely, a projective algebraic plane curve of homogeneous equation can be restricted to the affine algebraic plane curve of equation . These two operations are each inverse function, inverse to the other; therefore, the phrase algebraic plane curve is often used without specifying explicitly whether it is the affine or the projective case that is considered. If the defining polynomial of a plane algebraic curve is irreducible polynomial, irreducible, then one has an ''irreducible plane algebraic curve''. Otherwise, the algebraic curve is the union of one or several irreducible curves, called its ''Irreduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
