Projective Schemes and Proper Morphisms

Abstract

The notion of compactness is fundamental in topology and far beyond. For example, given a complex analytic space or manifold, one likes to construct a compact closure of it, a so-called compactification. Quite often suitable compactifications can make the original object more accessible.

There is a perfect adaptation of compactness to the scheme situation, the notion of properness that is dealt with in the present chapter. A relative scheme X⟶S is called proper if it is separated, of finite type, and universally closed. The latter means that the image of every closed subset of X is closed in S and, furthermore, that this behavior is not altered by base change on S. For example, the projective n-space $\mathbb{P}^{n}_{S}$ over some base scheme S is proper. In fact, the main theme of the present chapter is to characterize those proper S-schemes that are projective in the sense that they admit an immersion into a projective n-space over S. A thorough clarification of this problem is given by the theory of ample and very ample invertible sheaves, which is explained at length with all its prerequisites like Proj schemes, invertible sheaves, Serre twists, etc. Interpreting invertible sheaves in terms of divisors, the theory is applied to demonstrate that abelian varieties are projective.