View source for Complete variety

{{short description|Type of algebraic variety}}

In [[mathematics]], in particular in [[algebraic geometry]], a '''complete algebraic variety''' is an [[algebraic variety]]  {{mvar|X}}, such that for any variety {{mvar|Y}} the [[Product (category theory)|projection]] [[Regular map (algebraic geometry)|morphism]]

:<math>X \times Y \to Y</math>

is a [[closed map]] (i.e. maps [[closed set]]s onto closed sets).{{efn|Here the [[product (category theory)|product]] variety {{math|''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). See also [[Segre embedding]].}} This can be seen as an analogue of [[Compact space|compactness]] in algebraic geometry: a [[topological space]] {{mvar|X}} is compact if and only if the above projection map is closed with respect to topological products.

The image of a complete variety is closed and is a complete variety. A closed [[Algebraic_variety#Subvariety|subvariety]] of a complete variety is complete.

A complex variety is complete if and only if it is compact as a [[complex-analytic variety]].{{sfn|Mumford|1999|loc=Chapter I.10}}

The most common example of a complete variety is a [[projective variety]], but there do exist [[Hironaka's example|complete non-projective varieties]] in [[Dimension of an algebraic variety|dimensions]] 2 and higher. While any complete nonsingular surface is projective,<ref>{{cite journal |title=Introduction to the Problem of Minimal Models in the Theory of Algebraic Surfaces |first1=Oscar |last1=Zariski |journal=American Journal of Mathematics |volume=80 |year=1958 |pages=146–184 |doi=10.2307/2372827|jstor=2372827}}</ref> there exist nonsingular complete varieties in dimension 3 and higher which are not projective.<ref name="Nagata">{{cite journal |title=Existence theorems for nonprojective complete algebraic varieties |first1=Masayoshi |last1=Nagata |journal=Illinois J. Math. |volume=2 |year=1958 |pages=490–498 |doi=10.1215/ijm/1255454111|doi-access=free }}</ref> The first examples of non-projective complete varieties were given by [[Masayoshi Nagata]]<ref name="Nagata" /> and [[Heisuke Hironaka]].<ref>{{cite thesis |type=thesis |last=Hironaka |first=Heisuke |date=1960 |title=On the theory of birational blowing-up|publisher=Harvard University|url=https://id.lib.harvard.edu/alma/990038590670203941/catalog}}</ref> 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]].

== See also ==
*[[Chow's lemma]]
*[[Theorem of the cube]]
*[[Fano variety]]

==Notes==
{{notelist}}

==References==
{{reflist}}

==Sources==
{{refbegin}}
*Section II.4 of {{Hartshorne AG}}
*Chapter 7 of {{Citation
| last=Milne
| first=James S.
| title=Algebraic geometry
| year=2009
| version=v. 5.20
| url=http://jmilne.org/math/CourseNotes/ag.html
| accessdate=2010-08-04
}}
*Section I.9 of {{Citation
| last=Mumford
| first=David
| author-link=David Mumford
| title=The red book of varieties and schemes
| year=1999
| publisher=[[Springer-Verlag]]
| edition= Second, expanded
| doi=10.1007/b62130
| series=Lecture Notes in Mathematics
| volume=1358
| isbn=978-3-540-63293-1
}}
* {{eom|oldid=53623|title=Complete algebraic variety|first=V.I.|last= Danilov}}
*{{Cite book| last1=Hartshorne | first1=Robin | author1-link=Robin Hartshorne | title=Algebraic Geometry  | series=Graduate Texts in Mathematics | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-90244-9 | mr=0463157 | zbl=0367.14001 | year=1977 | volume=52 | url={{Google books|7z4mBQAAQBAJ|Algebraic Geometry|page=443|plainurl=yes}}|chapter=Appendix B. Example 3.4.1. (Fig.24) }}
{{refend}}

{{DEFAULTSORT:Complete Algebraic Variety}}
[[Category:Algebraic varieties]]