# 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.

## Key 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.