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

TL;DR

This paper introduces a new concept of L-regularity for coherent sheaves on the Grassmannian G(1,4), extending classical regularity notions, and uses it to classify certain low-rank vector bundles without inner cohomology.

Contribution

It defines L-regularity on G(1,4), proves classical property analogs, and classifies rank 2 and 3 vector bundles with no inner cohomology using monads.

Findings

Defined L-regularity for sheaves on G(1,4).

Proved splitting criterion for rank 2 bundles.

Classified rank 2 and 3 bundles without inner cohomology.

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