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Cone (algebraic Geometry)
In algebraic geometry, a cone is a generalization of a vector bundle. Specifically, given a scheme ''X'', the relative Spec :C = \operatorname_X R of a quasi-coherent graded ''O''''X''-algebra ''R'' is called the cone or affine cone of ''R''. Similarly, the relative Proj :\mathbb(C) = \operatorname_X R is called the projective cone of ''C'' or ''R''. Note: The cone comes with the \mathbb_m-action due to the grading of ''R''; this action is a part of the data of a cone (whence the terminology). Examples *If ''X'' = Spec ''k'' is a point and ''R'' is a homogeneous coordinate ring, then the affine cone of ''R'' is the (usual) affine cone over the projective variety corresponding to ''R''. *If R = \bigoplus_0^\infty I^n/I^ for some ideal sheaf ''I'', then \operatorname_X R is the normal cone to the closed scheme determined by ''I''. *If R = \bigoplus_0^\infty L^ for some line bundle ''L'', then \operatorname_X R is the total space of the dual of ''L''. *More generally, given a ve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne–Mumford Stack
In algebraic geometry, a Deligne–Mumford stack is a stack ''F'' such that Pierre Deligne and David Mumford introduced this notion in 1969 when they proved that moduli spaces of stable curves of fixed arithmetic genus are proper smooth Deligne–Mumford stacks. If the "étale" is weakened to " smooth", then such a stack is called an algebraic stack (also called an Artin stack, after Michael Artin). An algebraic space is Deligne–Mumford. A key fact about a Deligne–Mumford stack ''F'' is that any ''X'' in F(B), where ''B'' is quasi-compact, has only finitely many automorphisms. A Deligne–Mumford stack admits a presentation by a groupoid In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fu ...; see groupoid scheme. Examples Affine Stacks Deligne–Mumford stacks are typica ... [...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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor (algebraic Geometry)
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 subvarieti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weighted Projective Space
In algebraic geometry, a weighted projective space P(''a''0,...,''a''''n'') is the projective variety Proj(''k'' 'x''0,...,''x''''n'' associated to the graded ring ''k'' 'x''0,...,''x''''n''where the variable ''x''''k'' has degree ''a''''k''. Properties *If ''d'' is a positive integer then P(''a''0,''a''1,...,''a''''n'') is isomorphic to P(''da''0,''da''1,...,''da''''n''). This is a property of the Proj construction; geometrically it corresponds to the ''d''-tuple Veronese embedding. So without loss of generality one may assume that the degrees ''a''''i'' have no common factor. *Suppose that ''a''''0'',''a''''1'',...,''a''''n'' have no common factor, and that ''d'' is a common factor of all the ''a''i with ''i''≠''j'', then P(''a''0,''a''1,...,''a''''n'') is isomorphic to P(''a''0/d,...,''a''j-1/d,''a''j,''a''j+1/d,...,''a''''n''/d) (note that ''d'' is coprime to ''a''''j''; otherwise the isomorphism does not hold). So one may further assume that any set of ''n'' variables ''a'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological Line Bundle
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector subspace W \subseteq V, the fiber over W is the subspace W itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is :\mathcal_(-1), the dual of the hyperplane bundle or Serre's twisting sheaf \mathcal_(1). The hyp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane Bundle
In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k-dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector subspace W \subseteq V, the fiber over W is the subspace W itself. In the case of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since any vector bundle (over a compact space) is a pullback of the tautological bundle; this is to say a Grassmannian is a classifying space for vector bundles. Because of this, the tautological bundle is important in the study of characteristic classes. Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is :\mathcal_(-1), the dual of the hyperplane bundle or Serre's twisting sheaf \mathcal_(1). The hype ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Direct Limit
In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some directed set I, is denoted by \varinjlim A_i . This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms. Direct limits are a special case of the concept of colimit in category theory. Direct limits are dual to inverse limits, which are a special case of limits in category theory. Formal definition We will first give the definition for algebraic structures like groups and modules, and then the general definition, which can be used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmentation Map
In algebra, an augmentation of an associative algebra ''A'' over a commutative ring ''k'' is a ''k''-algebra homomorphism A \to k, typically denoted by ε. An algebra together with an augmentation is called an augmented algebra. The kernel of the augmentation is a two-sided ideal called the augmentation ideal of ''A''. For example, if A =k /math> is the group algebra of a finite group ''G'', then :A \to k,\, \sum a_i x_i \mapsto \sum a_i is an augmentation. If ''A'' is a graded algebra which is connected, i.e. A_0=k, then the homomorphism A\to k which maps an element to its homogeneous component of degree 0 is an augmentation. For example, :k to k, \sum a_ix^i \mapsto a_0 is an augmentation on the polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ... k /math>. Refe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Bundle
In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme ''X'' over a Noetherian scheme ''S'' is a P''n''-bundle if it is locally a projective ''n''-space; i.e., X \times_S U \simeq \mathbb^n_U and transition automorphisms are linear. Over a regular scheme ''S'' such as a smooth variety, every projective bundle is of the form \mathbb(E) for some vector bundle (locally free sheaf) ''E''. The projective bundle of a vector bundle Every vector bundle over a variety ''X'' gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way: there is an obstruction in the cohomology group ''H''2(''X'',O*). To see why, recall that a projective bundle comes equipped with transition functions on double intersections of a suitable open cover. On triple overlaps, any lift of these transition functions satisfies the cocycle condition up to an invertible function. The collection of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Spec
In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative 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 a sheaf of rings. 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 :\big\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows: For f\in 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 \big\ 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, \operatorname(R) is not, in general, a T1 space. However, \operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
