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 Pn-bundle if it...

tangent bundle of an intersection of spaces: Let Y 1 , Y 2 ⊂ X {\displaystyle Y_{1},Y_{2}\subset X} be projective subvarieties of a smooth projective variety...

group of line bundles on X {\displaystyle X} . Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because...

{\displaystyle X} into a projective space. A line bundle is ample if some positive power is very ample. An ample line bundle on a projective variety X {\displaystyle...

_{n}} is the P 1 {\displaystyle \mathbb {P} ^{1}} -bundle (a projective bundle) over the projective line P 1 {\displaystyle \mathbb {P} ^{1}} , associated...

of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since...

bundle comes from a divisor. (II) If X {\displaystyle X} is a projective scheme then the same statement holds. One of the most important line bundles...

bundle I-bundle Natural bundle Principal bundle Projective bundle Pullback bundle Quasifibration Universal bundle Vector bundle Wu–Yang dictionary Seifert...

standard round metric, the measure of projective space is exactly half the measure of the sphere. Real projective spaces are smooth manifolds. On Sn, in...

canonical bundle of a non-singular algebraic variety V {\displaystyle V} of dimension n {\displaystyle n} over a field is the line bundle Ω n = ω {\displaystyle...

In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in...

sheaf. The Euler sequence generalizes to that of a projective bundle as well as a Grassmann bundle (see the latter article for this generalization.) Let...

complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space...

{\displaystyle d} hypersurfaces of projective space P n {\displaystyle \mathbb {P} ^{n}} . This is given by the projective bundle H i l b d ( P n ) = P ( Γ (...

\mathbb {Z} _{2}} -bundle over S 1 {\displaystyle S^{1}} . Projective spaces provide some more interesting examples of principal bundles. Recall that the...

classifying spaces for vector bundle, among which projective spaces for line bundles Characteristic class Splitting principle Stable bundle Connection: the notion...

smooth projective complex algebraic variety, the category of representations of the fundamental group of the variety, and the category of Higgs bundles over...

concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet at infinity. A projective space may thus...

open subscheme of a projective space P A n {\displaystyle \mathbb {P} _{A}^{n}} over a ring A {\displaystyle A} . projective bundle If E is a locally free...

{\displaystyle |D|} is therefore a projective space. A linear system d {\displaystyle {\mathfrak {d}}} is then a projective subspace of a complete linear system...

written just as E, and the projective cone Proj X ⁡ R {\displaystyle \operatorname {Proj} _{X}R} is the projective bundle of E, which is written as P...

geometry, a line bundle on a projective variety is nef if it has nonnegative degree on every curve in the variety. The classes of nef line bundles are described...

fibers over real projective space RPn with fiber S0. The Hopf construction gives circle bundles p : S2n+1 → CPn over complex projective space. This is actually...

complex projective space, and that it is an example of the Eilenberg–Maclane space K ( Z , 2 ) . {\displaystyle K(\mathbb {Z} ,2).} Such bundles are classified...

schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental...

Narasimhan-Seshadri approach. If X is a smooth projective variety of dimension m and H is a hyperplane section, then a vector bundle (or a torsion-free sheaf) W is called...

using either Azumaya algebras over X or projective bundles over X. The second definition involves projective bundles that are locally trivial in the étale...

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...

the property of lifting that carries over from free to projective modules: a module P is projective if and only if for every surjective module homomorphism...

that the canonical bundle is a negative multiple of the ample line bundle O ( 1 ) {\displaystyle {\mathcal {O}}(1)} means that projective space is a Fano...