Barth, Hulek and Maruyama have showed that the moduli of stable rank 2 vector bundles on P2 are nonsingular rational varieties. There are also many examples of stable rank 2 vector bundles on P3. On the other hand, there is essentially only one example of rank 2 bundles on P4, which is constructed by Horrocks and Mumford. We hope the study of rank 2 bundles on hypersurfaces in P4 may give more insight to the study of vector bundles on P4. In this paper, we establish some general properties of stable rank 2 bundles on quadric hypersurfaces. We show the restriction theorem (1.4), (1.6), the existence of the spectrum (2.2), and the vanishing theorem (2.4), are also true for the stable rank 2 reflexive sheaves on quadric hypersurfaces just as in the case when the base variety is Pn. Though the methods to prove such results are similar to those we use for projective spaces, there are some technical difficulties. We should also mention that we shall always assume the base field is characteristic 0 and algebraically closed, and we shall use the definition of stability introduced by Mumford and Takemoto.
A lower bound for the number of components of the moduli schemes of stable rank 2 vector bundles on projective 3-folds
