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Euler Sequence
In mathematics, the Euler sequence is a particular exact sequence of sheaves on ''n''-dimensional projective space over a ring. It shows that the sheaf of relative differentials is stably isomorphic to an (n+1)-fold sum of the dual of the Serre twisting sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Statement Let \mathbb^n_A be the ''n''-dimensional projective space over a commutative ring ''A''. Let \Omega^1 = \Omega^1_ be the sheaf of 1-differentials on this space, and so on. The Euler sequence is the following exact sequence of sheaves on \mathbb^n_A: 0 \longrightarrow \Omega^1 \longrightarrow \mathcal(-1)^ \longrightarrow \mathcal \longrightarrow 0. The sequence can be constructed by defining a homomorphism S(-1)^ \to S, e_i \mapsto x_i with S = A _0, \ldots, x_n/math> and e_i = 1 in degree 1, surjective in degrees \geq 1, and checking that locally on the n+1 standard ch ... [...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] |
De Gruyter
Walter de Gruyter GmbH, known as De Gruyter (), is a German scholarly publishing house specializing in academic literature. History The roots of the company go back to 1749 when Frederick the Great granted the Königliche Realschule in Berlin the royal privilege to open a bookstore and "to publish good and useful books". In 1800, the store was taken over by Georg Reimer (1776–1842), operating as the ''Reimer'sche Buchhandlung'' from 1817, while the school’s press eventually became the ''Georg Reimer Verlag''. From 1816, Reimer used the representative Sacken'sche Palace on Berlin's Wilhelmstraße for his family and the publishing house, whereby the wings contained his print shop and press. The building became a meeting point for Berlin salon life and later served as the official residence of the president of Germany. Born in Ruhrort in 1862, Walter de Gruyter took a position with Reimer Verlag in 1894. By 1897, at the age of 35, he had become sole proprietor of the h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chow Group
In algebraic geometry, the Chow groups (named after Wei-Liang Chow by ) of an algebraic variety over any field are algebro-geometric analogs of the homology of a topological space. The elements of the Chow group are formed out of subvarieties (so-called algebraic cycles) in a similar way to how simplicial or cellular homology groups are formed out of subcomplexes. When the variety is smooth, the Chow groups can be interpreted as cohomology groups (compare Poincaré duality) and have a multiplication called the intersection product. The Chow groups carry rich information about an algebraic variety, and they are correspondingly hard to compute in general. Rational equivalence and Chow groups For what follows, define a variety over a field k to be an integral scheme of finite type over k. For any scheme X of finite type over k, an algebraic cycle on X means a finite linear combination of subvarieties of X with integer coefficients. (Here and below, subvarieties are understood to b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since found applications in physics, Calabi–Yau manifolds, string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants, topological quantum field theory, the Chern theorem etc. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Genus
In algebraic geometry, the geometric genus is a basic birational invariant of algebraic varieties and complex manifolds. Definition The geometric genus can be defined for non-singular complex projective varieties and more generally for complex manifolds as the Hodge number (equal to by Serre duality), that is, the dimension of the canonical linear system plus one. In other words for a variety of complex dimension it is the number of linearly independent holomorphic -forms to be found on .Danilov & Shokurov (1998), p. 53/ref> This definition, as the dimension of : then carries over to any base field, when is taken to be the sheaf of Kähler differentials and the power is the (top) exterior power, the canonical line bundle. The geometric genus is the first invariant of a sequence of invariants called the plurigenera. Case of curves In the case of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ample Line Bundle
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] |
Fano Varieties
In algebraic geometry, a Fano variety, introduced by Gino Fano in , is a complete variety ''X'' whose anticanonical bundle ''K''X* is ample. In this definition, one could assume that ''X'' is smooth over a field, but the minimal model program has also led to the study of Fano varieties with various types of singularities, such as terminal or klt singularities. Recently techniques in differential geometry have been applied to the study of Fano varieties over the complex numbers, and success has been found in constructing moduli spaces of Fano varieties and proving the existence of Kähler–Einstein metrics on them through the study of K-stability of Fano varieties. Examples * The fundamental example of Fano varieties are the projective spaces: the anticanonical line bundle of P''n'' over a field ''k'' is ''O''(''n''+1), which is very ample (over the complex numbers, its curvature is ''n+1'' times the Fubini–Study symplectic form). * Let ''D'' be a smooth codimension-1 subvarie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometry Of Projective Spaces
Projective space plays a central role in algebraic geometry. The aim of this article is to define the notion in terms of abstract algebraic geometry and to describe some basic uses of projective space. Homogeneous polynomial ideals Let k be an algebraically closed field (mathematics), field, and ''V'' be a finite-dimensional vector space over k. The symmetric algebra of the dual vector space ''V*'' is called the polynomial ring on ''V'' and denoted by k[''V'']. It is a naturally graded algebra by the degree of polynomials. The projective Nullstellensatz states that, for any homogeneous ideal ''I'' that does not contain all polynomials of a certain degree (referred to as an irrelevant ideal), the common zero locus of all polynomials in ''I'' (or ''Nullstelle'') is non-trivial (i.e. the common zero locus contains more than the single element ), and, more precisely, the ideal of polynomials that vanish on that locus coincides with the radical of an ideal, radical of the ideal ''I''. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Sheaf
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 divisor on ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Power
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Vector Field
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in many other branches of mathematics such as analytic number theory, complex analysis, and infinitesimal calculus. He introduced much of modern mathematical terminology and notation, including the notion of a mathematical function. He is also known for his work in mechanics, fluid dynamics, optics, astronomy and music theory. Euler is held to be one of the greatest mathematicians in history and the greatest of the 18th century. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Carl Friedrich Gauss remarked: "The study of Euler's works will remain the best school for the different fields of mathematics, and nothing else can replace it." Euler is als ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
