projective completion is with
: where is the degree of .

These two processes projective completion and restriction to an affine subspace are inverse one to the other. Therefore, an affine hypersurface and its projective completion have essentially the same properties, and are often considered as two points-of-view for the same hypersurface.

However, it may occur that an affine hypersurface is nonsingular, while its projective completion has singular points.

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