Main theorem of elimination theory: Difference between revisions

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Revision as of 02:50, 9 March 2018

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where <math>\overline\pi</math> is the prolongation of <math>\pi</math> to <math>L_x\times P_y.</math>

This is commonly expressed by saying the the origin of the affine plane is the projection of the point of the hyperbola that is at infinity, in the direction of the {{mvar|y}}-axis.

More generally, the image by <math>\pi</math> of every algebraic set in <math>L_x\times L_y</math> is either a finite number of points, or <math>L_x</math> with a finite number of points removed, while the image by <math>\overline\pi</math> of any algebraic set in <math>L_x\times P_y</math> is either a finite number of points or the whole line <math>L_y.</math> It follows that the image by <math>\overline\pi</math> of any algebraic set is an algebraic set, that is that <math>\overline\pi</math> is a closed map for Zariski topology.