Generically nef vector bundles on ruled surfaces

TL;DR

This paper establishes inequalities and bounds for generically nef vector bundles on ruled surfaces, extending known stability concepts and solving a problem posed by Peternell.

Contribution

It introduces Bogomolov type inequalities for generically nef vector bundles on ruled surfaces with various fiber restrictions, advancing the understanding of their invariants.

Findings

Proves a Bogomolov type inequality for nef vector bundles on ruled surfaces without negative sections.

Extends results to ruled surfaces with negative sections under certain conditions.

Provides bounds on invariants of curve fibrations factoring through ruled surfaces.

Abstract

The present paper concerns the invariants of generically nef vector bundles on ruled surfaces. By Mehta - Ramanathan Restriction Theorem and by Miyaoka characterization of semistable vector bundles on a curve, the generic nefness can be considered as a weak form of semistability. We establish a Bogomolov type inequality for generically nef vector bundles with nef general fiber restriction on ruled surfaces with no negative section. This gives an affermative answer in this case to a problem posed by Th. Peternell. Concerning ruled surfaces with a negative section, we prove a a similar result for generically nef vector bundles, with nef and balanced general fiber restriction and with a numerical condition on first Chern class, which is satisfied, for instance, if in its class there is a reduced divisor. Finally, we use such results to bound the invariants of curve fibrations, which factorize through finite covers of ruled surfaces.