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Grassmannian Bundle
In algebraic geometry, the Grassmann ''d''-plane bundle of a vector bundle ''E'' on an algebraic scheme ''X'' is a scheme over ''X'': :p: G_d(E) \to X such that the fiber p^(x) = G_d(E_x) is the Grassmannian of the ''d''-dimensional vector subspaces of E_x. For example, G_1(E) = \mathbb(E) is the projective bundle of ''E''. In the other direction, a Grassmann bundle is a special case of a (partial) flag bundle. Concretely, the Grassmann bundle can be constructed as a Quot scheme. Like the usual Grassmannian, the Grassmann bundle comes with natural vector bundles on it; namely, there are universal or tautological subbundle ''S'' and universal quotient bundle ''Q'' that fit into :0 \to S \to p^*E \to Q \to 0. Specifically, if ''V'' is in the fiber ''p''−1(''x''), then the fiber of ''S'' over ''V'' is ''V'' itself; thus, ''S'' has rank ''r'' = ''d'' = dim(''V'') and \wedge^d S is the determinant line bundle. Now, by the universal property of a projective bundle, the injection \wedge^r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Scheme
This is a glossary of algebraic geometry. See also glossary of commutative algebra, glossary of classical algebraic geometry, and glossary of ring theory. For the number-theoretic applications, see glossary of arithmetic and Diophantine geometry. For simplicity, a reference to the base scheme is often omitted; i.e., a scheme will be a scheme over some fixed base scheme ''S'' and a morphism an ''S''-morphism. !$@ A B C D E F G H I J K L M N O P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a field (mathematics), field K that has a differentiable structure. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V. When V is a real number, real or complex number, complex vector space, Grassmannians are compact space, compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsingular projective algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to \mathbf_2(\mathbf^4), parameterizing them by what are now called Plücker coordinates. (See below.) Herma ... [...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] |
Flag Bundle
In algebraic geometry, the flag bundle of a flagHere, E_i is a subbundle not subsheaf of E_. :E_: E = E_l \supsetneq \cdots \supsetneq E_1 \supsetneq 0 of vector bundles on an algebraic scheme ''X'' is the algebraic scheme over ''X'': :p: \operatorname(E_) \to X such that p^(x) is a flag V_ of vector spaces such that V_i is a vector subspace of (E_i)_x of dimension ''i''. If ''X'' is a point, then a flag bundle is a flag variety and if the length of the flag is one, then it is the Grassmann bundle In algebraic geometry, the Grassmann ''d''-plane bundle of a vector bundle ''E'' on an algebraic scheme ''X'' is a scheme over ''X'': :p: G_d(E) \to X such that the fiber p^(x) = G_d(E_x) is the Grassmannian of the ''d''-dimensional vector subspace ...; hence, a flag bundle is a common generalization of these two notions. Construction A flag bundle can be constructed inductively. References * *Expo. VI, § 4. of Algebraic geometry {{algebraic-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quot Scheme
In algebraic geometry, the Quot scheme is a scheme parametrizing sheaves on a projective scheme. More specifically, if ''X'' is a projective scheme over a Noetherian scheme ''S'' and if ''F'' is a coherent sheaf on ''X'', then there is a scheme \operatorname_F(X) whose set of ''T''-points \operatorname_F(X)(T) = \operatorname_S(T, \operatorname_F(X)) is the set of isomorphism classes of the quotients of F \times_S T that are flat over ''T''. The notion was introduced by Alexander Grothendieck. It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a Hilbert scheme. (In fact, taking ''F'' to be the structure sheaf \mathcal_X gives a Hilbert scheme.) Definition For a scheme of finite type X \to S over a Noetherian base scheme S, and a coherent sheaf \mathcal \in \text(X), there is a functor\mathcal_: (Sch/S)^ \to \textsending T \to S to\mathcal_(T) = \left\/ \simwhere X_T = X\times_ST and \mathcal_T = pr_X^*\mathcal under the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tautological Subbundle
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 hyper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant Line Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plane ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Plücker Embedding
In mathematics, the Plücker map embeds the Grassmannian \mathrm(k,V), whose elements are ''k''-Dimension (vector space), dimensional Linear subspace, subspaces of an ''n''-dimensional vector space ''V'', either real or complex, in a projective space, thereby realizing it as a projective algebraic variety. More precisely, the Plücker map Embedding, embeds \mathrm(k,V) into the projectivization \mathbb(^k V) of the k-th exterior power of V. The image is algebraic, consisting of the intersection of a number of quadrics defined by the (see below). The Plücker embedding was first defined by Julius Plücker in the case Plücker coordinates, k=2, n=4 as a way of describing the lines in three-dimensional space (which, as projective lines in real projective space, correspond to two-dimensional subspaces of a four-dimensional vector space). The image of that embedding is the Klein quadric in RP5. Hermann Grassmann generalized Plücker's embedding to arbitrary ''k'' and ''n''. The homo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relative Tangent Bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Second Fundamental Form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two"). Together with the first fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for a smooth immersed submanifold in a Riemannian manifold. Surface in R3 Motivation The second fundamental form of a parametric surface in was introduced and studied by Gauss. First suppose that the surface is the graph of a twice continuously differentiable function, , and that the plane is tangent to the surface at the origin. Then and its partial derivatives with respect to and vanish at (0,0). Therefore, the Taylor expansion of ''f'' at (0,0) starts with quadratic terms: : z=L\frac + Mxy + N\frac + \text\,, and the second fundamental form at the origin in the coordinates is the qua ... [...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 become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. 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 linearly independent sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
