Page "Projective variety" Paragraph 49

A projective variety is a complete variety ( i. e., proper over k ).

This is known as the main theorem of elimination theory.

Conversely, a complete variety is close to being a projective variety: Chow's lemma states that if X is a complete variety, there is a projective variety Z and a birational morphism Z → X.

( Moreover, through normalization, one can assume this projective variety is normal.