Classification of full exceptional collections of line bundles on three blow-ups of $\mathbb{P}^{3}$

Abstract

A fullness conjecture of Kuznetsov says that if a smooth projective variety $X$ admits a full exceptional collection of line bundles of length $l$, then any exceptional collection of line bundles of length $l$ is full. In this paper, we show that this conjecture holds for $X$ as the blow-up of $\mathbb{P}^{3}$ at a point, a line, or a twisted cubic curve, i.e. any exceptional collection of line bundles of length 6 on $X$ is full. Moreover, we obtain an explicit classification of full exceptional collections of line bundles on such $X$.Comment: 28 pages. To appear in Journal of the Korean Mathematical Society, A previous version with a different title appeared as [CGP17025] at https://cgp.ibs.re.kr/archive/preprints/201