Smooth blow up structures on projective bundles

TL;DR

This paper classifies certain projective bundles over projective spaces that admit smooth blow-up structures, providing new examples and extending understanding under Hartshorne's conjecture.

Contribution

It offers a classification of projective bundles with smooth blow-up structures over projective spaces, including new examples and generalizations beyond known cases.

Findings

Classified projective bundles over projective spaces with smooth blow-up structures.

Identified conditions under which vector bundles' projectivizations admit such structures.

Provided new examples of varieties with both projective bundle and smooth blow-up structures.

Abstract

Assuming Hartshorne's conjecture on complete intersections, we classify projective bundles over projective spaces which has a smooth blow up structure over another projective space. Under some assumptions, we also classify projective bundles over projective spaces which has a smooth blow up structure over some arbitrary smooth projective variety, not necessarily a projective space. We verify which of the globally generated vector bundles over projective space of first Chern class at most five has the property that their projectivisation has a smooth blow up structure, with no additional assumption. In the way, we get some new examples of varieties with both projective bundle and smooth blow up structures.