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Changing short description from "Theorem asserting that the image of a projective variety by a projection is also a variety" to "The image of a projective variety by a projection is also a variety" (Shortdesc helper) |
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:<math>(x,y)\mapsto \pi(x,y)=x.</math>
This projection is not [[closed map|closed]] for the [[Zariski topology]] (
the [[hyperbola]] {{mvar|H}} of equation <math>xy-1=0</math> is <math>L_x\setminus \{0\},</math> which is not closed, although {{mvar|H}} is closed, being an [[algebraic variety]].
If one extends <math>L_y</math> to a projective line <math>P_y,</math> the equation of the [[projective completion]] of the
:<math>xy_1-y_0=0,</math>
and contains
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==Geometrical interpretation==
In the preceding formulation, the [[polynomial ring]] <math>R[\mathbf x]=R[x_1, \ldots, x_n]</math> defines a [[morphism of schemes|morphism]] of [[Scheme (mathematics)|schemes]] (which are algebraic varieties if {{mvar|R}} is finitely generated over a field)
:<math>\mathbb{P}^{n-1}_R = \operatorname{Proj}(R[\mathbf x]) \to \operatorname{Spec}(R).</math>
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*{{cite book|last=Milne|first=James S.|title=The Abel Prize 2008–2012|chapter=The Work of John Tate|publisher=Springer|year=2014|isbn=9783642394492|author-link=James Milne (mathematician)}}
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