INTRODUCTION

Classically, two approaches have been proposed in algebraic surface theory: the first uses standard stuff on linear systems, adjunction and singularity theory. As usual, one gets results about some particular class of surfaces. The second uses a representation of the surface as a pencil of algebraic curves; that is, as a curve over a function field. This method is very useful for arithmetic applications. If the genus of the fibre is small, it can be used to describe certain classes of surfaces, such as elliptic surfaces and pencils of genus 2 (Xiao Gang), or to obtain some information on ‘atomic structure’ of surfaces, in Miles Reid's terminology.

Both of these methods use some geometric subobjects of the surface, such as curves and points. On the other hand, there exist also geometric objects lying over a surface, such as vector bundles and torsion-free sheaves, which are expressive enough to describe the geometry of the surface itself.

From a technical point of view, rank 2 vector bundles are extremely useful for understanding the underlying smooth structure of an algebraic surface. But I will try to convince you that they are also useful in algebraic geometry. To do this, let me recall two examples of the use of rank 2 vector bundles in the two approaches referred to above. In the first, any base point of a complete linear system determines a rank 2 vector bundle, whose geometry gives good information about this linear system.