Valuative Criteria for Families of Vector Bundles on Algebraic Varieties

Abstract

The object of this paper is to derive separation and completeness properties for the families of vector bundles on a nonsingular projective variety X over a fixed algebraically closed field k. By a vector bundle on X we mean a torsion-free coherent sheaf on X. If Xis a curve, such a sheaf must be locally free; thus this definition corresponds with the usual notion of vector bundle. On higher dimensional varieties, it appears that the category of locally free sheaves is too restrictive; for example, in this category, bundles do not in general have complete flags, whereas in the category of torsion-free sheaves, complete flags always exist (Proposition 1). Let S be an algebraic scheme over k. We define a family of vector bundles on X over S to be a coherent sheaf E on X x S, flat over S, such that for each s e S the induced sheaf E. on P`'(s) is torsion-free. We consider two such families E and E' to be equivalent if there is an invertible sheaf L on S such that E _ E' ( p*(L). Starting from the concept of a family of vector bundles, we are led to define a contravariant functor 'F3@ from the category of algebraic schemes over k to the category of sets, in the following way: If S is an algebraic scheme, let '0B,(S) be the set of equivalence classes of families of vector bundles on X over S. Then if T > S is a morphism of algebraic schemes and E a family of vector bundles over S, (g x X)*(E) will be a family of vector bundles over T. It is natural to ask whether the functor F3SX is representable; that is, is there an algebraic scheme V and a family of vector bundles E over V such that Hom (S, V) -= B,(S) for all S? Such a V would be a natural parameter space for the bundles on X, and E would be a universal family. It turns out that this is too much to expect. First of all, there are too many bundles to be parameterized by an algebraic scheme: we must break the functor up into separate parts corresponding to certain natural invariants, for example, the Hilbert polynomial. Next, we must throw away some