In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism
- X × Y → Y
is a closed map, i.e. maps closed sets onto closed sets.
The most common example of a complete variety is a projective variety, but there do exist complete and non-projective varieties in dimensions 2 and higher. The first examples of non-projective complete varieties were given by Masayoshi Nagata and Heisuke Hironaka. An affine space of positive dimension is not complete.
The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of 'complete', in the sense of 'no missing points', can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley.
Famous quotes containing the words complete and/or variety:
“Love is not enough. It must be the foundation, the cornerstone—but not the complete structure. It is much too pliable, too yielding.”
—Bette Davis (1908–1989)
“It has become a people’s war, and peoples of all sorts and races, of every degree of power and variety of fortune, are involved in its sweeping processes of change and settlement.”
—Woodrow Wilson (1856–1924)