Main theorem of elimination theory: Difference between revisions

Revision as of 21:23, 30 March 2019

Latest revision as of 01:49, 12 April 2025

(7 intermediate revisions by 4 users not shown)

Line 1:

{{short description|The image of a projective variety by a projection is also a variety}}

In [[algebraic geometry]], the '''main theorem of elimination theory''' states that every [[projective scheme]] is [[proper scheme|proper]]. A version of this theorem predates the existence of [[scheme theory]]. It can be stated, proved, and applied in the following more classical setting. Let {{math|''k''}} be a [[field (mathematics)|field]], denote by <math>\mathbb{P}_k^n</math> the {{math|''n''}}-dimensional [[projective space]] over {{math|''k''}}. The main theorem of elimination theory is the statement that for any {{math|''n''}} and any [[algebraic variety]] {{mvar|V}} defined over {{math|''k''}}, the projection map <math>V \times \mathbb{P}_k^n \to V</math> sends [[Zariski-closed]] subsets to Zariski-closed subsets.

Line 12 ⟶ 13:

:<math>(x,y)\mapsto \pi(x,y)=x.</math>

This projection is not [[closed map|closed]] for the [[Zariski topology]] (as well asnor for the usual topology if <math>k= \R</math> or <math>k= \C</math>), because the image by <math>\pi</math> of

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 parabolahyperbola becomes

:<math>xy_1-y_0=0,</math>

and contains

Line 54 ⟶ 55:

==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>

Line 69 ⟶ 70:

*{{cite book|last=Milne|first=James S.|title=The Abel Prize 2008&ndash;2012|chapter=The Work of John Tate|publisher=Springer|year=2014|isbn=9783642394492|author-link=James Milne (mathematician)}}

[[Category:AlgebraicTheorems in algebraic geometry]]