Se p 20 06 GEOMETRIC COLLECTIONS AND CASTELNUOVO - MUMFORD

The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf F on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to F and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties X with a geometric collection σ. We define the notion of regularity of a coherent sheaf F on X with respect to σ. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on P n and for a suitable geometric collection of coherent sheaves on P n both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface Q n ⊂ P n+1 (n odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo-Mumford regularity of their extension by zero in P n+1 .