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| In [[mathematics]], the '''Fulton–Hansen connectedness theorem''' is a result from [[intersection theory]] in [[algebraic geometry]], for the case of [[subvarieties]] of [[projective space]] with [[codimension]] large enough to make the intersection have components of dimension at least 1.
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| The formal statement is that if ''V'' and ''W'' are irreducible algebraic subvarieties of a [[projective space]] ''P'', all over an [[algebraically closed field]], and if
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| : dim(''V'') + dim (''W'') > dim (''P'')
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| in terms of the [[dimension of an algebraic variety]], then the intersection ''U'' of ''V'' and ''W'' is [[connected space|connected]].
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| More generally, the theorem states that if <math>Z</math> is a projective variety and <math>f:Z \to P^n \times P^n</math> is any morphism such that <math>\dim f(Z) > n</math>, then <math>f^{-1}\Delta</math> is connected, where <math>\Delta</math> is the [[diagonal]] in <math>P^n \times P^n</math>. The special case of intersections is recovered by taking <math>Z = V \times W</math>, with <math>f</math> the natural inclusion.
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| ==See also==
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| * [[Zariski's connectedness theorem]]
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| * [[Grothendieck's connectedness theorem]]
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| * [[Deligne's connectedness theorem]]
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| ==References==
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| * {{citation|first=W.|last= Fulton|first2= J. |last2=Hansen|title=A connectedness theorem for projective varieties with applications to intersections and singularities of mappings|journal= Annals of Math. |volume=110 |year=1979|pages= 159–166|doi=10.2307/1971249|jstor=1971249|issue=1|publisher=Annals of Mathematics}}
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| * {{citation|first=R.|last= Lazarsfeld|title=Positivity in Algebraic Geometry|publisher= Springer|year= 2004}}
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| ==External links==
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| * [http://www.math.unizh.ch/fileadmin/math/preprints/20-05.pdf PDF lectures withe the result as Theorem 15.3 (attributed to Faltings, also)]
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| {{DEFAULTSORT:Fulton-Hansen connectedness theorem}}
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| [[Category:Intersection theory]]
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| [[Category:Theorems in algebraic geometry]]
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The person who wrote the post is called Jayson Hirano and he totally digs that name. Distributing production is how he makes a living. Ohio is where my home is but my husband wants us to transfer. To climb is something I truly enjoy performing.
Here is my webpage: clairvoyants - go now -