Classification of full exceptional collections of line bundles on three blow-ups of $\mathbb{P}^{3}$
Wanmin Liu, Song Yang, Xun Yu
备注：
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/2017
发表信息：
Journal of the Korean Mathematical Society, Vol. 56, No. 2 (2019), 387-419
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 $X$.
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Phantom subcategories on blow-ups of Hirzebruch surfaces
School of Mathematics and Statistics, Guizhou University（贵州大学数学与统计学院）；School of Science, Hunan City University（湖南城市学院理学院）；School of Mathematics and Statistics, Changsha University of Science and Technology（长沙理工大学数学与统计学院）
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In the present paper, we focus on a weighted version of the Bounded Negativity Conjecture which predicts that for every smooth projective surface in characteristic zero the self-intersection numbers of reduced and irreducible curves are bounded from below by a global constant. We gather evidence for this conjecture by showing various bounds on the self-intersection number of curves in an algebraic surface. We focus our attention on blow-ups of algebraic surfaces, which have so far been neglected.
