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...

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

a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative"...

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...

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...

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...

real projective line, which is topologically equivalent to a circle. ⁠ R P 2 {\displaystyle \mathbb {RP} ^{2}} ⁠ is called the real projective plane...

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

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...

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

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...

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...

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

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

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...

{\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...

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...

n-dimensional linear system of divisors on a line bundle on X. The choice of a projective embedding of X, modulo projective transformations is likewise equivalent...

tangent bundle of projective space P n {\displaystyle \mathbb {P} ^{n}} over a field k {\displaystyle k} can be described in terms of the line bundle O (...

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...

especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action...

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...

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

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 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...

In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates...

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...