Main theorem of elimination theory: Difference between revisions

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

The '''main theorem of elimination theory''' states that a [[projective scheme]] is [[proper scheme|proper]].

== Sketch of proof ==

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Then <math>y \in Z_d \Leftrightarrow M(y)</math> has rank <math>< q \Leftrightarrow </math> all the <math>q \times q</math>-minors vanish at ''y''.

==See also==

*[[Elimination of quantifiers]]

*[[Resultant]]

*[[Gröbner basis]]

==References==