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 Pn. 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. [Formula: see text] for any i = 2,3,4) on G(1,4) by studying the associated monads.
A Cohomological Splitting Criterion for Rank 2 Vector Bundles on Hirzebruch Surfaces