Exceptional sequences of line bundles on projective bundles

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.

29 pages, several figures, expanded Section 6: there were some indices wrong in the statement of Theorem 6.1, these are now corrected and a proof added. Same content as published version