Complete algebraic variety

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