On cyclic strong exceptional collections of line bundles on surfaces
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On cyclic strong exceptional collections of line bundles on surfaces Alexey Elagin1 · Junyan Xu2 · Shizhuo Zhang3
Received: 22 November 2018 / Revised: 21 May 2020 / Accepted: 13 June 2020 © Springer Nature Switzerland AG 2020
Abstract We study exceptional collections of line bundles on surfaces. We prove that any full cyclic strong exceptional collection of line bundles on a rational surface is an augmentation in the sense of Lutz Hille and Markus Perling. We find simple geometric criteria of exceptionality (strong exceptionality, cyclic strong exceptionality) for collections of line bundles on weak del Pezzo surfaces. As a result, we classify smooth projective surfaces admitting a full cyclic strong exceptional collection of line bundles. Also, we provide an example of a weak del Pezzo surface of degree 2 and a full strong exceptional collection of line bundles on it which does not come from augmentations, thus answering a question by Hille and Perling. Keywords Weak del Pezzo surface · Exceptional collection · Line bundle Mathematics Subject Classification 14F05 · 14J26
Shizhuo Zhang is supported by ERC Consolidator grant WallCrossAG, no. 819864. Sections 2–5 were written by A. Elagin, while Sects. 6–8 and Appendix were written by J. Xu and S. Zhang. The work of A. Elagin is supported by the Russian Science Foundation under Grant 19-11-00164 and performed in Steklov Mathematical Institute of Russian Academy of Sciences.
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Shizhuo Zhang [email protected] Alexey Elagin [email protected] Junyan Xu [email protected]
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Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina St., Moscow, Russia 119991
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Department of Mathematics, Indiana University, 831 E. Third St., Bloomington, IN 47405, USA
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School of Mathematics, The University of Edinburgh, JCMB Building, Kings Building, Edinburgh EH9 3FD, UK
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1 Introduction Our paper is devoted to the study of exceptional collections of line bundles on surfaces. By a surface we always mean a smooth connected projective surface over an algebraically closed field of zero characteristic. Among the questions that are addressed in this paper are the following: Question 1.1 Which surfaces admit exceptional/strong exceptional/cyclic strong exceptional collections of line bundles? Question 1.2 How to construct exceptional/strong exceptional/cyclic strong exceptional collections of line bundles if they exist? Question 1.3 How to tell whether a given collection of line bundles on a surface is exceptional/strong exceptional/cyclic strong exceptional? It is believed that any variety with a full exceptional collection in the bounded derived category of coherent sheaves is rational. Though, there is no proof yet even in the case when the collection is formed by line bundles. On the other hand, on any rational surface one can construct a full exceptional collection of line bundles, using a construction by Dmitry Orlov [21,23]. For strong exceptional collections of line bundles the question is much more complicated. First, it is know
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