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:
“A complete woman is probably not a very admirable creature. She is manipulative, uses other people to get her own way, and works within whatever system she is in.”
—Anita Brookner (b. 1938)
“The catalogue of forms is endless: until every shape has found its city, new cities will continue to be born. When the forms exhaust their variety and come apart, the end of cities begins.”
—Italo Calvino (1923–1985)