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A fullness conjecture of Kuznetsov says that if a smooth projective variety$X$ admits a full exceptional collection of line bundles of length $l$, thenany exceptional collection of line bundles of length $l$ is full. In thispaper, 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. anyexceptional collection of line bundles of length 6 on $X$ is full. Moreover, weobtain an explicit classification of full exceptional collections of linebundles on such $X$.

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