Editing Projective variety (section)

== Relation to complete varieties ==

By definition, a variety is [[complete variety|complete]], if it is [[proper map|proper]] over ''k''. The [[valuative criterion of properness]] expresses the intuition that in a proper variety, there are no points "missing".

There is a close relation between complete and projective varieties: on the one hand, projective space and therefore any projective variety is complete. The converse is not true in general. However:

*A [[Singularity theory#Algebraic curve singularities|smooth curve]] ''C'' is projective if and only if it is [[Complete variety|complete]]. This is proved by identifying ''C'' with the set of [[discrete valuation ring]]s of the [[function field of an algebraic variety|function field]] ''k''(''C'') over ''k''. This set has a natural Zariski topology called the [[Zariski–Riemann space]].
* [[Chow's lemma]] states that for any complete variety ''X'', there is a projective variety ''Z'' and a [[Birational geometry#Birational maps 2|birational morphism]] ''Z'' → ''X''.<ref>{{harvnb|Grothendieck|Dieudonné|1961|loc=5.6}}</ref> (Moreover, through [[normal variety|normalization]], one can assume this projective variety is normal.)

Some properties of a projective variety follow from completeness. For example,

:<math>\Gamma(X, \mathcal{O}_X) = k</math>

for any projective variety ''X'' over ''k''.<ref>{{harvnb|Hartshorne|1977|loc=Ch II. Exercise 4.5}}</ref> This fact is an algebraic analogue of [[Liouville's theorem (complex analysis)|Liouville's theorem]] (any holomorphic function on a connected compact complex manifold is constant). In fact, the similarity between complex analytic geometry and algebraic geometry on complex projective varieties goes much further than this, as is explained below.

[[Quasi-projective variety|Quasi-projective varieties]] are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes [[affine variety|affine varieties]]. Affine varieties are almost never complete (or projective). In fact, a projective subvariety of an affine variety must have dimension zero. This is because only the constants are globally [[regular function]]s on a projective variety.