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

Wanmin Liu|Song Yang|Xun Yu
대한수학회 (The Korean Mathematical Society)|2019.03|대한수학회지 (Journal of the KMS)|제56권 제2호|pp. 387-419|DOI 10.4134/JKMS.j180204

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참고문헌

키워드

derived category of coherent sheavesfull exceptional collectionsemiorthogonal decomposition
주제분류수학

초록

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 $X$.

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 $X$.

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