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

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 ${\mathbf{P}}^{\mathbf{n}}$. 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. $H_{\ast }^i(E)=H^i(E\otimes \text{Q})=0$ for any $i=2,3,4$) on G(1,4) by studying the associated monads.