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 Classification of full exceptional collections of line bundles on three blow-ups of $\mathbb{P}^{3}$ J. Korean Math. Soc. 2019 Vol. 56, No. 2, 387-419 Published online 2019 Mar 01 Wanmin Liu, Song Yang, Xun Yu Uppsala University, Tianjin University; Tianjin University 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 $X$. Keywords : derived category of coherent sheaves, full exceptional collection, semiorthogonal decomposition MSC numbers : 14F05, 14J45, 18E30 Full-Text :