 Complete algebraic variety

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.^{[1]}
The most common example of a complete variety is a projective variety, but there do exist complete and nonprojective varieties in dimensions 2 and higher. The first examples of nonprojective complete varieties were given by Masayoshi Nagata^{[2]} and Heisuke Hironaka.^{[citation needed]} 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.
Notes
 ^ Here the product variety X × Y does not carry the product topology, in general; the Zariski topology on it will have more closed sets (except in very simple cases).
 ^ Existence theorems for nonprojective complete algebraic varieties, Illinois J. Math. 2 (1958) 490–498.
References
 Section II.4 of Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: SpringerVerlag, ISBN 9780387902449, MR0463157
 Chapter 7 of Milne, James S. (2009), Algebraic geometry, v. 5.20, http://jmilne.org/math/CourseNotes/ag.html, retrieved 20100804
 Section I.9 of Mumford, David (1999), The red book of varieties and schemes, Lecture notes in mathematics, 1358 (Second, expanded ed.), SpringerVerlag, doi:10.1007/b62130, ISBN 9783540632931
Categories: Algebraic varieties
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