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A generalization of the concept of a compact complex algebraic variety. A separated variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237101.png" /> is called complete if for any variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237102.png" /> the projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237103.png" /> is a closed morphism, i.e. it maps closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237104.png" /> (in the Zariski topology) into closed subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237105.png" />. There is what is called the valuative completeness criterion: For any discrete valuation ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237106.png" /> with field of fractions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237107.png" /> and any morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237108.png" /> there should be a unique morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c0237109.png" /> that extends <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371010.png" />. This condition is an analogue of the requirement that any sequence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371011.png" /> has a limit point.
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A generalization of the concept of a compact complex algebraic variety. A separated variety $X$ is called complete if for any variety $Y$ the projection $X \times Y \rightarrow Y$  is a closed morphism, i.e. it maps closed subsets of $X \times Y$ (in the [[Zariski topology]]) into closed subsets of $Y$.  
  
Any projective variety is complete, but not vice versa. For any complete algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371012.png" /> there exists a projective variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371013.png" /> and a projective birational morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371014.png" /> (Chow's lemma). For any algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371015.png" /> there exists an open imbedding into a complete variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c023/c023710/c02371016.png" /> (Nagata's theorem). A generalization of the concept of a complete algebraic variety to the relative case is that of a proper morphism of schemes.
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Any [[projective variety]] is complete, but not vice versa. For any complete algebraic variety $X$ there exists a projective variety $X_1$ and a projective birational morphism $X_1\rightarrow X$ (Chow's lemma). For any algebraic variety $X$ there exists an open imbedding into a complete variety $\tilde X$ (Nagata's theorem). A generalization of the concept of a complete algebraic variety to the relative case is that of a [[proper morphism]] of schemes.
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There is also the ''valuative completeness criterion'': For any discrete valuation ring $A$ with [[field of fractions]] $K$ and any morphism $u : \mathrm{Spec}\,K \rightarrow X$ there should be a unique morphism $v : \mathrm{Spec}\,A \rightarrow X$ that extends $v$. This condition is an analogue of the requirement that any sequence in $X$ has a limit point.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne,   "Algebraic geometry" , Springer (1977)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich,   "Basic algebraic geometry" , Springer (1977) (Translated from Russian)</TD></TR></table>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top"> R. Hartshorne, "Algebraic geometry" , Springer (1977) {{ISBN|0-387-90244-9}} {{MR|0463157}} {{ZBL|0367.14001}} </TD></TR>
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<TR><TD valign="top">[2]</TD> <TD valign="top"> I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) {{MR|0447223}} {{ZBL|0362.14001}} </TD></TR>
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</table>
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{{TEX|done}}

Latest revision as of 19:12, 24 November 2023

A generalization of the concept of a compact complex algebraic variety. A separated variety $X$ is called complete if for any variety $Y$ the projection $X \times Y \rightarrow Y$ is a closed morphism, i.e. it maps closed subsets of $X \times Y$ (in the Zariski topology) into closed subsets of $Y$.

Any projective variety is complete, but not vice versa. For any complete algebraic variety $X$ there exists a projective variety $X_1$ and a projective birational morphism $X_1\rightarrow X$ (Chow's lemma). For any algebraic variety $X$ there exists an open imbedding into a complete variety $\tilde X$ (Nagata's theorem). A generalization of the concept of a complete algebraic variety to the relative case is that of a proper morphism of schemes.

There is also the valuative completeness criterion: For any discrete valuation ring $A$ with field of fractions $K$ and any morphism $u : \mathrm{Spec}\,K \rightarrow X$ there should be a unique morphism $v : \mathrm{Spec}\,A \rightarrow X$ that extends $v$. This condition is an analogue of the requirement that any sequence in $X$ has a limit point.

References

[1] R. Hartshorne, "Algebraic geometry" , Springer (1977) ISBN 0-387-90244-9 MR0463157 Zbl 0367.14001
[2] I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001
How to Cite This Entry:
Complete algebraic variety. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Complete_algebraic_variety&oldid=11651
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article