P1-bundles over projective manifolds of Picard number one each of which admit another smooth morphism of relative dimension one

TL;DR

This paper classifies P1-bundles over projective manifolds with Picard number one that also admit an additional smooth morphism of relative dimension one, providing a comprehensive understanding of their structure.

Contribution

It offers a complete classification of such P1-bundles, revealing their structure and the conditions under which they admit another smooth morphism.

Findings

Classification of P1-bundles with additional smooth morphism

Conditions for the existence of a second smooth morphism

Structural insights into projective manifolds of Picard number one

Abstract

We give a complete classification of P1-bundles over a projective manifold of Picard number one which admit another smooth morphism of relative dimension one.