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On exceptional collections of line bundles on toric deligne-mumford stacks
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Title
On exceptional collections of line bundles on toric deligne-mumford stacks
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Wang
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Chengxi
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Wang, Chengxi
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author
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Borisov
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Lev
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Lev Borisov
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Advisory Committee
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chair
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Buch
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Anders Buch
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internal member
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Gibney
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Angela
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Angela Gibney
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internal member
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de Jong
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Johan
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Johan de Jong
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Advisory Committee
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Rutgers University
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School of Graduate Studies
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theses
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2020
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2020-05
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English
Abstract
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We study strong exceptional collections of line bundles on Fano toric Deligne-Mumford stacks $mathbb{P}_{mathbf{Sigma}}$. We prove that when the rank of Picard group is no more than two, any strong exceptional collection of line bundles generates the derived category of $mathbb{P}_{mathbf{Sigma}}$, as long as the number of elements in the collection equals the rank of the (Grothendieck) $K$-theory group of $mathbb{P}_{mathbf{Sigma}}$.
Moreover, we consider generalized Hirzebruch surfaces $mathbb{F}_{alpha,n}$ which are not Fano and have Picard rank two. We give a classification of all (strong) exceptional collections of line bundles of maximum length and show they generate the derived category, which is a generalization for the results of Hirzebruch surfaces. We show that any exceptional collections of line bundles on $mathbb{F}_{alpha,n}$ can be extend to maximum length $2(alpha+1)$ which is the rank of $K$-theory.
We give examples of strong exceptional collections of line bundles on $mathbb{F}_{alpha,n}$ which cannot be extended to strong exceptional collections of line bundles of length $2(alpha+1)$, but can be extend to exceptional collections of line bundles of maximum length $2(alpha+1)$.
Moreover, we consider generalized Hirzebruch surfaces $mathbb{F}_{alpha,n}$ which are not Fano and have Picard rank two. We give a classification of all (strong) exceptional collections of line bundles of maximum length and show they generate the derived category, which is a generalization for the results of Hirzebruch surfaces. We show that any exceptional collections of line bundles on $mathbb{F}_{alpha,n}$ can be extend to maximum length $2(alpha+1)$ which is the rank of $K$-theory.
We give examples of strong exceptional collections of line bundles on $mathbb{F}_{alpha,n}$ which cannot be extended to strong exceptional collections of line bundles of length $2(alpha+1)$, but can be extend to exceptional collections of line bundles of maximum length $2(alpha+1)$.
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Mathematics
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Rutgers University Electronic Theses and Dissertations
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School of Graduate Studies Electronic Theses and Dissertations
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rucore10001600001
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ETD_10605
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doi:10.7282/t3-6388-cs26
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1 online resource (vi, 64 pages)
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Ph.D.
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Includes bibliographical references
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ETD doctoral
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The author owns the copyright to this work.
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Wang
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Chengxi
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Permission or license
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2020-03-12 10:06:46
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Chengxi Wang
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Rutgers University. School of Graduate Studies
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Author Agreement License
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I hereby grant to the Rutgers University Libraries and to my school the non-exclusive right to archive, reproduce and distribute my thesis or dissertation, in whole or in part, and/or my abstract, in whole or in part, in and from an electronic format, subject to the release date subsequently stipulated in this submittal form and approved by my school. I represent and stipulate that the thesis or dissertation and its abstract are my original work, that they do not infringe or violate any rights of others, and that I make these grants as the sole owner of the rights to my thesis or dissertation and its abstract. I represent that I have obtained written permissions, when necessary, from the owner(s) of each third party copyrighted matter to be included in my thesis or dissertation and will supply copies of such upon request by my school. I acknowledge that RU ETD and my school will not distribute my thesis or dissertation or its abstract if, in their reasonable judgment, they believe all such rights have not been secured. I acknowledge that I retain ownership rights to the copyright of my work. I also retain the right to use all or part of this thesis or dissertation in future works, such as articles or books.
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2020-05-31
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2022-05-31
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Access to this PDF has been restricted at the author's request. It will be publicly available after May 31st, 2022.
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Copyright protected
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Open
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