• Journal of the Korean Mathematical Society
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• v.56 no.2
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• pp.387-419
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• 2019

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.

• Bulletin of the Korean Mathematical Society
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• v.54 no.5
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• pp.1827-1849
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• 2017

We prove that if a smooth projective algebraic variety of dimension less or equal to three has a unit type integral K-motive, then its integral Chow motive is of Lefschetz type. As a consequence, the integral Chow motive is of Lefschetz type for a smooth projective variety of dimension less or equal to three that admits a full exceptional collection.