Low Rank Vector Bundles on the Grassmannian G(1,4)

Abstract

Here we define the concept of L-regularity for coherent sheaves on the Grassmannian G(1, 4) as a generalization of Castelnuovo-Mumford regularity on P n . In this setting we prove analogs of some classical properties. We use our notion of L-regularity in order to prove a splitting criterion for rank 2 vector bundles with only a finite number of vanishing conditions. In the second part we give the classification of rank 2 and rank 3 vector bundles without "inner" cohomology (i.e. H i * (E) = H i (E ⊗ Q) = 0 for any i = 2, 3, 4) on G(1, 4) by studying the associated monads.