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:
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She that has that is clad in complete steel,
And like a quivered nymph with arrows keen
May trace huge forests and unharbored heaths,
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—John Milton (1608–1674)
“To invent without scruple a new principle to every new phenomenon, instead of adapting it to the old; to overload our hypothesis with a variety of this kind, are certain proofs that none of these principles is the just one, and that we only desire, by a number of falsehoods, to cover our ignorance of the truth.”
—David Hume (1711–1776)