About: Grassmann bundle

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In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: such that the fiber is the Grassmannian of the d-dimensional vector subspaces of . For example, 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 . , ,

Attributes	Values
rdfs:label	Grassmann bundle (en)
rdfs:comment	In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: such that the fiber is the Grassmannian of the d-dimensional vector subspaces of . For example, 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 . , , (en)
dcterms:subject	Algebraic geometry
Wikipage page ID	50406093 (xsd:integer)
Wikipage revision ID	1117316324 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective bundle Determinant line bundle Grassmannian Ergebnisse der Mathematik und ihrer Grenzgebiete Algebraic geometry Chern class Plücker embedding Second fundamental form Euler sequence Quot scheme Flag bundle Springer-Verlag Tautological subbundle Relative tangent bundle Algebraic scheme
sameAs	Grassmann bundle Grassmann bundle
dbp:wikiPageUsesTemplate	dbt:Citation dbt:Reflist
has abstract	In algebraic geometry, the Grassmann d-plane bundle of a vector bundle E on an algebraic scheme X is a scheme over X: such that the fiber is the Grassmannian of the d-dimensional vector subspaces of . For example, 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 . 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 is the determinant line bundle. Now, by the universal property of a projective bundle, the injection corresponds to the morphism over X: , which is nothing but a family of Plücker embeddings. The relative tangent bundle TGd(E)/X of Gd(E) is given by which morally is given by the second fundamental form. In the case d = 1, it is given as follows: if V is a finite-dimensional vector space, then for each line in V passing through the origin (a point of ), there is the natural identification (see Chern class#Complex projective space for example): and the above is the family-version of this identification. (The general care is a generalization of this.) In the case d = 1, the early exact sequence tensored with the dual of S = O(-1) gives: , which is the relative version of the Euler sequence. (en)
prov:wasDerivedFrom	wikipedia-en:Grassmann_bundle?oldid=1117316324&ns=0
page length (characters) of wiki page	2899 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Grassmann_bundle
is Link from a Wikipage to another Wikipage of	Projective bundle Gauss map Grassmannian Localized Chern class Tautological bundle Grassmannian bundle Euler sequence List of things named after Hermann Grassmann Flag bundle Segre class
is Wikipage redirect of	Grassmannian bundle
is foaf:primaryTopic of	wikipedia-en:Grassmann_bundle

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