- Complete algebraic variety
- X × Y → Y
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.
- Section II.4 of Hartshorne, Robin (1977), Algebraic Geometry, Graduate Texts in Mathematics, 52, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR0463157
- Chapter 7 of Milne, James S. (2009), Algebraic geometry, v. 5.20, http://jmilne.org/math/CourseNotes/ag.html, retrieved 2010-08-04
- Section I.9 of Mumford, David (1999), The red book of varieties and schemes, Lecture notes in mathematics, 1358 (Second, expanded ed.), Springer-Verlag, doi:10.1007/b62130, ISBN 978-3-540-63293-1
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