It is based on a variant of the so-called Standard Construction which is explained in Sect. The Sects. We close Sect. Finally, we study, in Sect. By a pointwe always mean a closed point. The composite morphism:. The sequence:. This morphism is called the second fundamental form of the exact sequence 2. Then the composite morphism.

Assume that such a morphism is not generically zero.

One can easily show that. Then one has an exact sequence:. One has an exact sequence:. The result of O. In this particular case, the arguments used by Benoist become substantially simpler. One has an isomorphism:. Keeping the notation from par. One derives an exact sequence:.

From 3.

### No document with DOI "10.1.1.218.457"

We described in par. It follows that:. One deduces that:. We put:. Consider the incidence diagram from par. As we saw in par. The first assertion from the second point of the conclusion is now clear.

The next result, in which we use the method described in Sect. It induces a diagram:. We have seen in par. Dualizing the exact sequence 5. One sees easily that the image of the composite morphism:.

**Determine whether vectors span R3 and is the collection a basis?**

The approach to the Standard Construction used in Sect. Assume the claim, for the moment.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. MathOverflow is a question and answer site for professional mathematicians.

It only takes a minute to sign up. Since I triple checked my computations, I would like to know if the multiplicative property of the Chern classes on the exact sequences does not hold here.

OK, I'll write my comment as an answer. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Restriction of vector bundles Ask Question. Asked 2 years, 1 month ago. Active 2 years, 1 month ago. Viewed times. Thanks in advance.

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### A refined stable restriction theorem for vector bundles on quadric threefolds

Sign up using Facebook. Sign up using Email and Password.Fix a line bundle L of degree d on Y. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Bhosle Usha N, Generalised parabolic bundles and applications to torsion free sheaves on nodal curves, Arkiv for Matematik 30 2 — Bhosle Usha N, Picard groups of moduli of vector bundles, Math. Indian Acad. Sci Math. Bhosle Usha N, Maximal subsheaves of torsion free sheaves on nodal curves, J.

London Math. Ein L and Lazarsfeld R, Stability and restrictions of Picard bundles with an application to the normal bundles of elliptic curves, Complex projective geometry, edited by G Ellingsrud et al.

Li Y, Spectral curves, theta divisors and Picard bundles, Int. Newstead P E Introduction to moduli problems and orbit spaces. Springer: Berlin. Google Scholar. Ramanujan — a tribute, Studies in Math. Rego C J, Compactification of the space of vector bundles on a singular curve, Comm. Sun X, Degeneration of moduli spaces and generalised theta functions, J. Algebraic Geom. Download references. The author would like to thank Indranil Biswas for useful discussions. Correspondence to Usha N Bhosle.

Reprints and Permissions. Bhosle, U. Picard bundle on the moduli space of torsionfree sheaves. Proc Math Sci34 Download citation. Received : 14 March Revised : 21 January Accepted : 29 January Published : 06 June Search SpringerLink Search. Immediate online access to all issues from Subscription will auto renew annually.The uniform rank-2 vector bundles on P n are determined and the behaviour of the stable rank-2 vector bundles on P 2 under restriction to a general line is studied, where P n denotes the n-dimensional projective space over an algebraically closed field of positive characteristic.

This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Barth, W. Hartshorne, R. Lange, H. Maruyama, M. Akizuki, Tokyo Sato, E. Japan 28, Download references.

Reprints and Permissions. On stable and uniform rank-2 vector bundles on P 2 in characteristic p. Manuscripta Math 29, 11—28 Download citation. Received : 12 January Issue Date : March Search SpringerLink Search. Abstract The uniform rank-2 vector bundles on P n are determined and the behaviour of the stable rank-2 vector bundles on P 2 under restriction to a general line is studied, where P n denotes the n-dimensional projective space over an algebraically closed field of positive characteristic.

Immediate online access to all issues from Subscription will auto renew annually. Taxes to be calculated in checkout. References [1] Barth, W. Akizuki, Tokyo [5] Sato, E. Japan 28, [6] van de Ven, A.In this work we deal with vector bundles of rank two on a Fano manifold X with second and fourth Betti numbers equal to one. We study the nef and pseudoeffective cones of the corresponding projectivizations and how these cones are related to the decomposability of the vector bundle.

As consequences, we obtain the complete list of P 1 -bundles over X that have a second P 1 -bundle structure, classify all the uniform rank two vector bundles on this class of Fano manifolds, and show the stability of indecomposable Fano bundles with one exception on P 2.

Source Kyoto J. Zentralblatt MATH identifier Kyoto J. More by Luis E. Abstract Article info and citation First page References Abstract In this work we deal with vector bundles of rank two on a Fano manifold X with second and fourth Betti numbers equal to one. Article information Source Kyoto J.

Export citation. Export Cancel. References [APW] V. Ancona, T. Peternell, and J. You have access to this content. You have partial access to this content.

You do not have access to this content. More like this.In mathematicsthe orthogonal group in dimension ndenoted O nis the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal groupby analogy with the general linear group. The orthogonal group is an algebraic group and a Lie group. It is compact.

The orthogonal group in dimension n has two connected components. The one that contains the identity element is a subgroup, called the special orthogonal groupand denoted SO n. It consists of all orthogonal matrices of determinant 1. This group is also called the rotation groupgeneralizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point in dimension 2 or a line in dimension 3.

In the other connected component all orthogonal matrices have —1 as a determinant.

More generally, given a non-degenerate symmetric bilinear form or quadratic form [1] on a vector space over a fieldthe orthogonal group of the form is the group of invertible linear maps that preserve the form.

The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot productor, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groupssince the condition of preserving a form can be expressed as an equality of matrices.

The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space E of dimension nthe elements of the orthogonal group O n are, up to a uniform scaling homothecythe linear maps from E to E that map orthogonal vectors to orthogonal vectors. Let E n be the group of the Euclidean isometries of a Euclidean space S of dimension n.

This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. This stabilizer is or, more exactly, is isomorphic to O nsince the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space. There is a natural group homomorphism p from E n to O nwhich is defined by. The kernel of p is the vector space of the translations. So, the translation form a normal subgroup of E nthe stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to O n.

Moreover, the Euclidean group is a semidirect product of O n and the group of translations.In mathematicsa 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 spacea manifoldor 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.

The simplest example is the case that the family of vector spaces is constant, i. 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 manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles.

For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial. Vector bundles are almost always required to be locally trivialhowever, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle respectively.

Complex vector bundles can be viewed as real vector bundles with additional structure.

## Orthogonal group

In the following, we focus on real vector bundles in the category of topological spaces. If k x is equal to a constant k on all of Xthen k is called the rank of the vector bundle, and E is said to be a vector bundle of rank k. Often the definition of a vector bundle includes that the rank is well defined, so that k x is constant. Vector bundles of rank 1 are called line bundleswhile those of rank 2 are less commonly called plane bundles. These are called the transition functions or the coordinate transformations of the vector bundle.

This is sometimes taken as the definition of a vector bundle. The class of all vector bundles together with bundle morphisms forms a category. Restricting to vector bundles for which the spaces are manifolds and the bundle projections are smooth maps and smooth bundle morphisms we obtain the category of smooth vector bundles.

Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundlesand are also often called vector bundle homomorphisms. A bundle homomorphism from E 1 to E 2 with an inverse which is also a bundle homomorphism from E 2 to E 1 is called a vector bundle isomorphismand then E 1 and E 2 are said to be isomorphic vector bundles. An isomorphism of a rank k vector bundle E over X with the trivial bundle of rank k over X is called a trivialization of Eand E is then said to be trivial or trivializable.

The definition of a vector bundle shows that any vector bundle is locally trivial.

## thoughts on “Rank 2 spanned vector bundles on p 2 with a fixed restriction to a”