Introduction

In this paper I try to show how the exceptional bundles can be useful to study vector bundles on projective spaces. The exceptional bundles appeared in [5], and they were used to describe the ranks and Chern classes of semi-stable sheaves. In [1] the generalized Beilinson spectral sequence, built with exceptional bundles, was defined, and it was used in [2] and [3] to describe some moduli spaces of semi-stable sheaves on ℙ2. The general notion of exceptional bundle and helix, on ℙn and many other varieties, is due mainly to A.L. Gorodentsev and A.N. Rudakov (cf. [7], [14]). A.N. Rudakov described completely in [12] the exceptional bundles on ℙ1 X ℙ1, and used them in [13] to describe the ranks and Chern classes of semi-stable sheaves on this variety. The exceptional vector bundles on ℙ3 have been studied (cf [4], [10], [11]) but they have not yet been used to describe semi-stable sheaves on ℙ3. On higher ℙn almost nothing is known.

In the second part of this paper, new invariants of coherent sheaves of non-zero rank are defined. In some cases they are more convenient than the Chern classes.

In the third part the exceptional bundles and helices are defined, and their basic properties are given.

In the fourth part, I define some useful hypersurfaces in the space of invariants of coherent sheaves on ℙn.