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| In [[mathematics]], the '''tautological bundle''' is a particular natural [[vector bundle]] occurring over a [[Grassmannian]]: each Grassmannian has a single tautological bundle. In the case of [[projective space]] this is known as the ''[[tautological line bundle]].'' The older term ''canonical bundle'' has dropped out of favour, on the grounds that ''[[canonical (disambiguation)|canonical]]'' is heavily overloaded as it is, in mathematical terminology, and (worse) confusion with the [[canonical class]] in [[algebraic geometry]] could scarcely be avoided.
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| Grassmannians by definition are the parameter spaces for [[linear subspace]]s, of a given dimension, in a given [[vector space]] ''W''. If ''G'' is a Grassmannian, and ''V''<sub>''g''</sub> is the subspace of ''W'' corresponding to ''g'' in ''G'', this is already almost the data required for a vector bundle: namely a vector space for each point ''g'', varying continuously. All that can stop the definition of the tautological bundle from this indication, is the (pedantic) difficulty that the ''V''<sub>''g''</sub> are going to intersect. Fixing this up is a routine application of the [[disjoint union]] device, so that the bundle projection is from a [[Fiber bundle|total space]] made up of identical copies of the ''V''<sub>''g''</sub>, that now do not intersect. With this, we have the bundle.
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| The projective space case is included: see [[tautological line bundle]]. By convention and use ''P''(''V'') may usefully carry the tautological bundle in the [[dual space]] sense. That is, with ''V''<sup>*</sup> the dual space, points of ''P''(''V'') carry the vector subspaces of ''V''<sup>*</sup> that are their kernels, when considered as (rays of) [[linear functional]]s on ''V''<sup>*</sup>. If ''V'' has dimension ''n'' + 1, the tautological [[line bundle]] is one tautological bundle, and the other, just described, is of rank ''n''.
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| ==Properties==
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| * The [[Picard group]] of line bundles on <math>\mathbb{P}(V)</math> is [[infinite cyclic]], and the [[tautological line bundle]] is a generator.
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| * In the case of projective space, where the tautological bundle is a [[line bundle]], the associated [[invertible sheaf]] of sections is <math>\mathcal{O}(-1)</math>, the tensor inverse (''ie'' the dual vector bundle) of the hyperplane bundle or [[Proj#The twisting sheaf|Serre twist sheaf]] <math>\mathcal{O}(1)</math>; in other words the hyperplane bundle is the generator of the Picard group having positive degree (as a [[divisor (algebraic geometry)|divisor]]) and the tautological bundle is its opposite: the generator of negative degree.
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| ==See also==
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| * [[Hopf bundle]]
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| ==References==
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| *{{Citation | last1=Griffiths | first1=Phillip | author1-link=Phillip Griffiths | last2=Harris | first2=Joseph | author2-link=Joe Harris (mathematician) | title=Principles of algebraic geometry | publisher=[[John Wiley & Sons]] | location=New York | series=Wiley Classics Library | isbn=978-0-471-05059-9 | id={{MathSciNet | id = 1288523}} | year=1994}}.
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| *{{Citation | last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=[[Algebraic Geometry (book)|Algebraic Geometry]] | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | oclc=13348052 | id={{MathSciNet | id = 0463157}} | year=1977}}.
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| [[Category:Vector bundles]]
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