Exceptional sequences of line bundles on projective bundles

TL;DR

This paper classifies exceptional line bundle sequences on projective bundles, especially for cotangent bundles of projective spaces, and confirms a conjecture relating Rouquier dimension to the dimension of the total space.

Contribution

It provides a complete classification of exceptional sequences for certain projective bundles and proves the Rouquier dimension equals the dimension of the total space for general cases.

Findings

Complete classification for $ ext{dim}( ext{base})=2$

Any maximal exceptional sequence is full in this case

Rouquier dimension equals the dimension of the total space for general $ ext{dim}( ext{base})$

Abstract

For a vector bundle $\mathcal E \to \mathbb P^\ell$ we investigate exceptional sequences of line bundles on the total space of the projectivisation $X = \mathbb P(\mathcal E)$. In particular, we consider the case of the cotangent bundle of $\mathbb P^\ell$. If $\ell = 2$, we completely classify the (strong) exceptional sequences and show that any maximal exceptional sequence is full. For general $\ell$, we prove that the Rouquier dimension of $\mathcal D(X)$ equals $\dim X$, thereby confirming a conjecture of Orlov.