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:
“Seldom, very seldom, does complete truth belong to any human disclosure; seldom can it happen that something is not a little disguised, or a little mistaken.”
—Jane Austen (1775–1817)
“A poem is like a person. Though it has a family tree, it is important not because of its ancestors but because of its individuality. The poem, like any human being, is something more than its most complete analysis. Like any human being, it gives a sense of unified individuality which no summary of its qualities can reproduce; and at the same time a sense of variety which is beyond satisfactory final analysis.”
—Donald Stauffer (b. 1930)