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| {{expert-subject|mathematics|date=November 2012}}
| | Emilia Shryock is my name but you can contact me something you like. South Dakota is where me and my husband reside and my family members enjoys it. To do aerobics is a factor that I'm totally addicted to. I am a meter reader.<br><br>My web blog - [http://www.cam4teens.com/user/MWolff http://www.cam4teens.com/user/MWolff] |
| {{about|the general Plücker embedding|the classical case of 2-planes in 4-space|Plücker coordinates}}
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| In mathematics, the '''Plücker embedding''' describes a method to realize the [[Grassmannian]] of all ''r''-dimensional [[Linear subspace|subspace]]s of an ''n''-dimensional [[vector space]] ''V'' as a [[subvariety]] of the [[projective space]] of the ''r''th [[exterior power]] of that vector space, '''P'''(∧<sup>''r''</sup> ''V'').
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| The Plücker embedding was first defined, in the case ''r'' = 2, ''n'' = 4, in [[Plücker coordinates|coordinates]] by [[Julius Plücker]] as a way of describing the lines in three dimensional space (which, as [[projective line]]s in real projective space, correspond to two dimensional subspaces of a four dimensional vector space). This was generalized by [[Hermann Grassmann]] to arbitrary ''r'' and ''n'' using a generalization of Plücker's coordinates, sometimes called '''Grassmann coordinates'''.
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| == Definition ==
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| The Plücker embedding (over the field ''K'') is the map ''ι'' defined by
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| :<math>
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| \begin{align}
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| \iota \colon \mathbf{Gr}(r, K^n) &{}\rightarrow \mathbf{P}(\wedge^r K^n)\\
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| \operatorname{span}( v_1, \ldots, v_r ) &{}\mapsto K( v_1 \wedge \cdots \wedge v_r )
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| \end{align}
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| </math>
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| where '''Gr'''(''r'', ''K''<sup>''n''</sup>) is the [[Grassmannian]], i.e., the space of all ''r''-dimensional subspaces of the ''n''-dimensional [[vector space]], ''K''<sup>''n''</sup>.
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| This is an isomorphism from the Grassmannian to the image of ''ι'', which is a projective variety. This variety can be completely characterized as an intersection of quadrics, each coming from a relation on the Plücker (or Grassmann) coordinates that derives from linear algebra.
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| The [[bracket ring]] appears as the ring of polynomial functions on the exterior power.<ref>{{citation | last1=Björner | first1=Anders | last2=Las Vergnas |author2-link=Michel Las Vergnas| first2=Michel | last3=Sturmfels | first3=Bernd | author3-link=Bernd Sturmfels | last4=White | first4=Neil | last5=Ziegler | first5=Günter | title=Oriented matroids | edition=2nd | series=Encyclopedia of Mathematics and Its Applications | volume=46 | publisher=[[Cambridge University Press]] | year=1999 | isbn=0-521-77750-X | zbl=0944.52006 | page=79 }}</ref>
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| ==References==
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| {{reflist}}
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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 | edition=2nd | isbn=0-471-05059-8 | mr=1288523 | year=1994 | zbl=0836.14001 }}
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| ==Further reading==
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| * {{ cite book | last1=Miller | first1=Ezra | last2=Sturmfels | first2=Bernd | author2-link=Bernd Sturmfels | title=Combinatorial commutative algebra | series=Graduate Texts in Mathematics | volume=227 | location=New York, NY | publisher=[[Springer-Verlag]] | isbn=0-387-23707-0 | year=2005 | zbl=1090.13001 }}
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| {{DEFAULTSORT:Plucker embedding}}
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| [[Category:Algebraic geometry]]
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| [[Category:Differential geometry]]
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Emilia Shryock is my name but you can contact me something you like. South Dakota is where me and my husband reside and my family members enjoys it. To do aerobics is a factor that I'm totally addicted to. I am a meter reader.
My web blog - http://www.cam4teens.com/user/MWolff