Castelnuovo-Mumford Regularity over Scrolls and Splitting Criteria

TL;DR

This paper extends Castelnuovo-Mumford regularity to scrolls formed from line bundles on projective spaces and establishes splitting criteria for vector bundles, broadening understanding of bundle behavior in complex geometric contexts.

Contribution

It introduces a generalized regularity concept for scrolls and proves Horrocks-type splitting criteria for vector bundles on these varieties.

Findings

Generalized Castelnuovo-Mumford regularity for scrolls.

Proved Horrocks-type splitting criteria for vector bundles.

Established natural extension of regularity from projective spaces.

Abstract

We introduce and study a notion of Castelnuovo-Mumford regularity suitable for scrolls obtained as projectivisations of sums of line bundles on $\mathbb P^m$. We show that this is a natural generalisation of the well known regularity on projective and multiprojective spaces and we prove Horrocks-type splitting criteria for vector bundles.