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Fiber bundle whose fibers are projective spaces
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
Projective_bundle
Branch of mathematics
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
K-theory
Vector bundle existing over a Grassmannian
of projective space the tautological bundle is known as the tautological line bundle. The tautological bundle is also called the universal bundle since
Tautological_bundle
Analogs of homology groups for algebraic varieties
group of line bundles on X {\displaystyle X} . Rationally equivalent cycles defined by hypersurfaces are easy to construct on projective space because
Chow_group
Ruled surface over the projective line
_{n}} is the P 1 {\displaystyle \mathbb {P} ^{1}} -bundle (a projective bundle) over the projective line P 1 {\displaystyle \mathbb {P} ^{1}} , associated
Hirzebruch_surface
Concept in algebraic geometry
{\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
Ample_line_bundle
Mathematical concept
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
Complex_projective_space
Short exact sequence of sheaves on projective space
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
Euler_sequence
Vector bundle of rank 1
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
Line_bundle
Continuous surjection satisfying a local triviality condition
bundle I-bundle Natural bundle Principal bundle Projective bundle Pullback bundle Quasifibration Universal bundle Vector bundle Wu–Yang dictionary Seifert
Fiber_bundle
Algebraic variety in a 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
Projective_variety
Fiber bundle whose fibers are group torsors
\mathbb {Z} _{2}} -bundle over S 1 {\displaystyle S^{1}} . Projective spaces provide some more interesting examples of principal bundles. Recall that the
Principal_bundle
bundle theorem. The bundle theorem is analogous for Möbius planes to the Theorem of Desargues for projective planes. From the bundle theorem follows the
Bundle_theorem
Geometric space whose points represent algebro-geometric objects of some fixed kind
{\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 (
Moduli_space
Type of topological space
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
Real_projective_space
Generalization of a vector bundle
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
Cone_(algebraic_geometry)
Bundle of linear subspaces of the tangent bundle
bundle is obtained by combining Grassmannians of the tangent spaces at each point, it is a special case of the Grassmann bundle and of the projective
Contact_bundle
Concept in algebraic geometry
that of projective curves. Here, the canonical bundle is the same as the (holomorphic) cotangent bundle. A global section of the canonical bundle is therefore
Canonical_bundle
Type of vector bundle
smooth projective complex algebraic variety, the category of representations of the fundamental group of the variety, and the category of Higgs bundles over
Higgs_bundle
Projective analogue of the spectrum of a ring
schemes, which produces objects with the typical properties of projective spaces and projective varieties. The construction, while not functorial, is a fundamental
Proj_construction
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
Grassmann_bundle
Concept in algebraic geometry
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
Nef_line_bundle
Mathematical parametrization of vector spaces by another space
classifying spaces for vector bundle, among which projective spaces for line bundles Characteristic class Splitting principle Stable bundle Connection: the notion
Vector_bundle
Scheme parametrizing flags in the fibers of a vector bundle
chooses a line in a quotient bundle. Thus, the complete flag bundle is a tower of projective bundles: Fl ( E ) ⟶ Fl 1 , … , n − 2 ( E ) ⟶ ⋯ ⟶ P ( E ) ⟶
Flag_bundle
Completion of the usual space with "points at infinity"
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
Projective_space
generalized to projective connections by Michael Eastwood et al. in Tractor bundles can be defined for arbitrary parabolic geometries. The tractor bundle for a
Tractor_bundle
Abelian group related to division algebras
using either Azumaya algebras over X or projective bundles over X. The second definition involves projective bundles that are locally trivial in the étale
Brauer_group
Bogomolov, Thomas Bridgeland and many others. On a smooth projective variety, line bundles of given numerical invariants are parametrised over a well-behaved
Stable_vector_bundle
algebraic geometry, the Horrocks–Mumford bundle is an indecomposable rank 2 vector bundle on 4-dimensional projective space P4 introduced by Geoffrey Horrocks
Horrocks–Mumford_bundle
Fiber bundle of the 3-sphere over the 2-sphere, with 1-spheres as fibers
known as Hopf fibrations. First, one can replace the projective line by an n-dimensional projective space. Second, one can replace the complex numbers by
Hopf_fibration
Concept in mathematics
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates
Quaternionic_projective_space
Method for constructing vector bundles
Horrocks construction is a method for constructing vector bundles, especially over projective spaces, introduced by Geoffrey Horrocks (1964, section 10)
Horrocks_construction
Quotient of special unitary group by its center
isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projective space. In terms of matrices
Projective_unitary_group
Digital storefront company selling video games and e-books
Humble Bundle, Inc. is a digital storefront for video games, which grew out of its original offering of Humble Bundles, collections of games sold at a
Humble_Bundle
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
Algebraic geometry of projective spaces
Algebraic_geometry_of_projective_spaces
exceptional line of a vector bundle over projective space is a projective line in projective space where the vector bundle has exceptional behavior, in
Jumping_line
Direct summand of a free module (mathematics)
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
Projective_module
Type of fiber bundle on a Riemann surface
indigenous bundle on a Riemann surface is a fiber bundle with a flat connection associated to some complex projective structure. Indigenous bundles were introduced
Indigenous_bundle
Complex vector bundle on a complex manifold
vector bundles on a smooth complex projective variety X (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e.
Holomorphic_vector_bundle
geometry, a stable principal bundle is a generalisation of the notion of a stable vector bundle to the setting of principal bundles. The concept of stability
Stable_principal_bundle
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
Quot_scheme
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
Glossary of algebraic geometry
Glossary_of_algebraic_geometry
Indian university teacher (born 1971)
fundamental properties, such as the contravariant functoriality and a projective bundle formula, as well as constructing an action of the usual higher Chow
Amalendu_Krishna
Construction in group theory
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
Projective_linear_group
algebraic geometry, a Tango bundle is one of the indecomposable vector bundles of rank n − 1 constructed on n-dimensional projective space Pn by Tango (1976)
Tango_bundle
Generalization of vector bundles
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 (
Coherent_sheaf
Relates the geometric vector bundles to algebraic projective modules
of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules are
Serre–Swan_theorem
Classifies holomorphic vector bundles over the complex projective line
classifies holomorphic vector bundles over the complex projective line. In particular every holomorphic vector bundle over C P 1 {\displaystyle \mathbb
Birkhoff–Grothendieck_theorem
Mathematical object studied in the field of algebraic geometry
moduli of nice objects tend not to be projective but only quasi-projective. Another case is a moduli of vector bundles on a curve. Here, there are the notions
Algebraic_variety
Principal fiber bundle
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
Circle_bundle
of convex varieties are projective bundles P ( E ) {\displaystyle \mathbb {P} ({\mathcal {E}})} for an algebraic vector bundle E → C {\displaystyle {\mathcal
Convexity (algebraic geometry)
Convexity_(algebraic_geometry)
Concept in algebraic geometry
{\displaystyle |D|} is therefore a projective space. A linear system d {\displaystyle {\mathfrak {d}}} is then a projective subspace of a complete linear system
Linear_system_of_divisors
Surface containing a line through every point
surface). Every minimal projective ruled surface other than the projective plane is the projective bundle of a 2-dimensional vector bundle over some curve. The
Ruled_surface
Mathematical space
Grassmannian was by Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to G r 2 ( R 4 ) {\displaystyle
Grassmannian
canonical bundle K. The 0th graded component R 0 {\displaystyle R_{0}} is sections of the trivial bundle, and is one-dimensional as V is projective. The projective
Canonical_ring
Fictional character by Agatha Christie
Lady Eileen "Bundle" Brent is a fictional character of two of the Agatha Christie novels, The Secret of Chimneys (1925) and The Seven Dials Mystery (1929)
Bundle_Brent
Type of transport in differential geometry
having the same unparametrized geodesics. Projective connections are modeled on the geometry of projective space. In modern terms, they may be described
Projective_connection
Characteristic classes of vector bundles
characteristic classes for projective space forms the basis for many characteristic class computations since for any smooth projective subvariety X ⊂ P n {\displaystyle
Chern_class
Field of algebraic geometry
determine whether two smooth projective varieties are birational. A projective variety X is called minimal if the canonical bundle KX is nef. For X of dimension
Birational_geometry
Vector bundles theorem
holomorphic (or algebraic) vector bundles over compact Riemann surfaces (or non-singular projective algebraic curves), to projective unitary representations of
Kobayashi–Hitchin correspondence
Kobayashi–Hitchin_correspondence
Problem in algebraic geometry
homomorphism. Let E be a vector bundle on X of rank r and q: P(E ⊕ 1) → X the projective bundle (here 1 means the trivial line bundle). As usual, we identity
Residual_intersection
Generalizations of codimension-1 subvarieties of algebraic varieties
If X is a projective curve over k, then the divisor of a nonzero rational function f on X has degree zero. As a result, for a projective curve X, the
Divisor_(algebraic_geometry)
Improved version of the vi text editor
as VimL), but can be written in other languages as well. There are projects bundling together complex scripts and customizations and aimed at turning Vim
Vim_(text_editor)
Result in algebraic geometry
Chern class) on a smooth projective curve over a field k {\displaystyle k} has a formula similar to Riemann–Roch for line bundles. If we take X = C {\displaystyle
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Relation between genus, degree, and dimension of function spaces over surfaces
where C is a projective non-singular algebraic curve over an algebraically closed field k. In fact, the same formula holds for projective curves over any
Riemann–Roch_theorem
Mathematical technique for vector bundles
Grothendieck splitting principle for holomorphic vector bundles on the complex projective line H. Blane Lawson and Marie-Louise Michelsohn, Spin Geometry
Splitting_principle
and quotient bundles. With E = Sym 2 ( S ∗ ⊗ Q ∗ ) {\displaystyle E=\operatorname {Sym} ^{2}(S^{*}\otimes Q^{*})} , the projective bundle q : X = P (
Segre_class
Model of the extended complex plane plus a point at infinity
manifolds. In projective geometry, the sphere is an example of a complex projective space and can be thought of as the complex projective line P 1 ( C
Riemann_sphere
Riemannian manifold with SU(n) holonomy
algebraic variety embedded in a projective space is a Kähler manifold, because there is a natural Fubini–Study metric on a projective space which one can restrict
Calabi–Yau_manifold
Study of vector bundles, principal bundles, and fibre bundles
between solutions to the self-duality equations and algebraic bundles over the complex projective space C P 3 {\displaystyle \mathbb {CP} ^{3}} . Another significant
Gauge_theory_(mathematics)
In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space V = (V
Projective_orthogonal_group
Concept in algebraic geometry
of smooth projective varieties X. That is, this vector space is canonically identified with the corresponding space for any smooth projective variety which
Kodaira_dimension
Projective variety that is also an algebraic group
analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law
Abelian_variety
Type of directory bundle
descendants macOS, iOS, iPadOS, tvOS, watchOS, and visionOS, and in GNUstep, a bundle is a file directory with a defined structure and file extension, allowing
Bundle_(macOS)
Theorem in algebraic geometry
proved by Jean-Pierre Serre. The basic version applies to vector bundles on a smooth projective variety, but Alexander Grothendieck found wide generalizations
Serre_duality
gives conditions for a line bundle on a projective surface to be very ample. Let D be a nef divisor on a smooth projective surface X. Denote by KX the
Reider's_theorem
Possibility of a consistent definition of "clockwise" in a mathematical space
planes, and tori are orientable, for example. But Möbius strips, real projective planes, and Klein bottles are non-orientable. They, as visualized in 3
Orientability
Gives general conditions under which sheaf cohomology groups with indices > 0 are zero
Positivity of the line bundle L translates into the corresponding invertible sheaf being ample (i.e., some tensor power gives a projective embedding). The algebraic
Kodaira_vanishing_theorem
Correspondsnce between Higgs bundles and fundamental group representations
Simpson) is a correspondence between Higgs bundles and representations of the fundamental group of a smooth, projective complex algebraic variety, or a compact
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
if L is a big nef line bundle (for example, an ample line bundle) on a complex projective manifold with canonical line bundle K, then the coherent cohomology
Kawamata–Viehweg vanishing theorem
Kawamata–Viehweg_vanishing_theorem
scheme admitting an ample family of line bundles, as opposed to an ample line bundle. In particular, a quasi-projective variety is a divisorial scheme and the
Divisorial_scheme
Fiber bundle Principal bundle Frame bundle Hopf bundle Associated bundle Vector bundle Tangent bundle Cotangent bundle Line bundle Jet bundle Sheaf (mathematics)
List of differential geometry topics
List_of_differential_geometry_topics
Vector bundle of cotangent spaces at every point in a manifold
mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold
Cotangent_bundle
commutative ring assigned to any projective variety. If V is an algebraic variety given as a subvariety of projective space of a given dimension N, its
Homogeneous_coordinate_ring
American video streaming service
also announced a bundle including its other U.S. streaming services Hulu (ad-supported version) and ESPN+, marketed as The Disney Bundle, initially for
Disney+
Number used in algebraic geometry
In mathematics, the degree of an affine or projective variety of dimension n is the number of intersection points of the variety with n hyperplanes in
Degree of an algebraic variety
Degree_of_an_algebraic_variety
Television station in Pittsburgh
and started Project Bundle Up, an operation to make sure that children and seniors receive warm clothing. WTAE-TV has run the Project Bundle Up Auction
WTAE-TV
Concept in differential geometry
reasons; see below.) The complex projective plane CP2 is not spin. More generally, all even-dimensional complex projective spaces CP2n are not spin. All
Spin_structure
Concept in algebraic geometry
leads to many numerical invariants for projective varieties. For example, if X {\displaystyle X} is a smooth projective curve over an algebraically closed
Coherent_sheaf_cohomology
Topological space
even this is homeomorphic to the projective plane times the circle, otherwise it is homeomorphic to a surface bundle associated to an orientation reversing
Seifert_fiber_space
Manifold with Riemannian, complex and symplectic structure
automatically projective varieties. Shing-Tung Yau proved the Calabi conjecture: every smooth projective variety with ample canonical bundle has a Kähler–Einstein
Kähler_manifold
becomes a projective space bundle (the Picard bundle). It has been studied in detail, for example by Kempf and Mukai. Let C be a smooth projective curve of
Symmetric product of an algebraic curve
Symmetric_product_of_an_algebraic_curve
defines the d-dimensional embedding of X over a splitting field L. Projective bundle Jacobson (1996), p. 113 Gille & Szamuely (2006), p. 129 Gille & Szamuely
Severi–Brauer_variety
integers j are the line bundles on projective space. Full exceptional collections have also been constructed on all smooth projective toric varieties, del
Semiorthogonal_decomposition
Concept in algebraic geometry
del Pezzo surface is either a product of two projective lines (with d=8), or the blow-up of a projective plane in 9 − d points with no three collinear
Del_Pezzo_surface
Manifold
varieties are complex manifolds, including: Complex vector spaces. Complex projective spaces, Pn(C). Complex Grassmannians. Complex Lie groups such as GL(n
Complex_manifold
embedding of X over S. The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing
Cotangent_sheaf
Mathematical concept
greatest common divisor equal to 1, and L is some line bundle on the smooth curve S. If S is projective (or equivalently, compact), then the degree of L is
Elliptic_surface
Construction in differential topology
differential topology, the jet bundle is a certain construction that makes a new smooth fiber bundle out of a given smooth fiber bundle. It makes it possible to
Jet_bundle
Iitaka dimension of a line bundle L on an algebraic variety X is the dimension of the image of the rational map to projective space determined by L. This
Iitaka_dimension
PROJECTIVE BUNDLE
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