Dedicated to Peter Newstead for his 65th anniversary

Abstract

We construct Beilinson type spectral sequences on scrolls and apply them to vector bundles on rational scrolls. The first application is a cohomological criterion for a vector bundle to be globally generated, Corollary 4.7. The relative canonical bundle can also be described by cohomological conditions, Proposition 4.8.

Introduction

In recent years, the theory of derived categories received considerable attention from the mathematical community. Remarkable works have been done in the attempt to understand the way the derived categories reflect the geometry of varieties. In some particular cases (projective bundles, Grassmannianns, quadrics etc), the derived categories have been described explicitly. In other cases, the description can be reduced to some known derived categories. For example, it was shown in [Orl] that if we control the derived category of a projective variety X, then we can control the derived category of any projective bundle on X. Orlov's result relies on a relative Beilinson spectral sequence obtained from a resolution of the diagonal inside the fibered product [Orl, p. 855–856].

The aim of this note is to construct slightly different Beilinson type sequences on scrolls, using resolutions of diagonals inside the usual product, see Section 4. Working with the usual product instead of the fibered-product has the advantage of giving information on the vanishing of Hochschild cohomology, [Ca]. The precise relationship between resolutions of diagonals and Beilinson type spectral sequences is recalled in Section 3.