Castelnuovo-Mumford Regularity and Splitting Criteria for Logarithmic Bundles over Rational Normal Scroll Surfaces

TL;DR

This paper develops a new notion of Castelnuovo-Mumford regularity for rational normal scroll surfaces, establishing splitting criteria, characterizing Ulrich bundles, and analyzing logarithmic bundles related to line and rational curve arrangements.

Contribution

It introduces a regularity concept for scroll surfaces and provides new splitting criteria and characterizations for Ulrich bundles in this context.

Findings

Established analogs of classical properties for the new regularity notion.

Derived splitting criteria for coherent sheaves on scroll surfaces.

Characterized Ulrich bundles and studied logarithmic bundles for line and rational curve arrangements.

Abstract

We introduce and study a notion of Castelnuovo-Mumford regularity suitable for rational normal scroll surfaces. In this setting we prove analogs of some classical properties. We prove splitting criteria for coherent sheaves and a characterization of Ulrich bundles. Finally we study logarithmic bundles associated to arrangements of lines and rational curves.