If is the twisting sheaf of Serre on , we let denote the pullback of to ; that is, for the canonical map A scheme X → S is called projective over S if it factors as a closed immersion followed by the projection to S. A line bundle (or invertible sheaf) on a scheme X over S is said to be very ample relative to S if there is an immersion (i.e., an open immersion followed by a closed immersion) for some n so that pullbacks to . Then a S-scheme X is projective if and only if it is proper and there exists a very ample sheaf on X relative to S. Indeed, if X is proper, then an immersion corresponding to the very ample line bundle is necessarily closed. Conversely, if X is projective, then the pullback of under the closed immersion of X into a projective space is very ample. That "projective" implies "proper" is deeper: the main theorem of elimination theory. Relation to complete varieties By definition, a variety is complete, if it is 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 smooth curve C is projective if and only if it is complete. This is proved by identifying C with the set of discrete valuation rings of the 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 morphism Z → X. (Moreover, through normalization, one can assume this projective variety is normal.) Some properties of a projective variety follow from completeness. For example, for any projective variety X over k. This fact is an algebraic analogue of 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 varieties are, by definition, those which are open subvarieties of projective varieties. This class of varieties includes affine varieties. Affine varieties are almost never complete (or projective). In fact,