Main theorem of elimination theory: Difference between revisions

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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|''mk''}}, the projection map <math>\mathbb{P}_k^nV \times \mathbb{P}_k^mn \to \mathbb{P}_k^mV</math> sends [[Zariski-closed]] subsets to Zariski-closed subsets. Since Zariski-closed subsets in projective spaces are related to [[homogeneous polynomial]]s, it's possible to state the theorem in that language directly, as was customary in the [[elimination theory]].

===A simple motivating example===