Karmazyn, J., Kuznetsov, A. and Shinder, E. (Submitted: 2019) Derived categories of singular surfaces. arXiv. (Submitted)
Abstract
We develop an approach that allows to construct semiorthogonal decompositions of derived categories of surfaces with cyclic quotient singularities whose components are equivalent to derived categories of local finite dimensional algebras. We first explain how to induce a semiorthogonal decomposition of a surface X with rational singularities from a semiorthogonal decomposition of its resolution. In the case when X has cyclic quotient singularities, we introduce the condition of adherence for the components of the semiorthogonal decomposition of the resolution that allows to identify the components of the induced decomposition with derived categories of local finite dimensional algebras. Further, we present an obstruction in the Brauer group of X to the existence of such semiorthogonal decomposition, and show that in the presence of the obstruction a suitable modification of the adherence condition gives a semiorthogonal decomposition of the twisted derived category of X. We illustrate the theory by exhibiting a semiorthogonal decomposition for the untwisted or twisted derived category of any normal projective toric surface depending on whether its Weil divisor class group is torsion-free or not. For weighted projective planes we compute the generators of the components explicitly and relate our results to the results of Kawamata based on iterated extensions of reflexive sheaves of rank 1.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2019 The Author(s). For reuse permissions, please contact the Author(s). |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 26 Mar 2019 11:12 |
Last Modified: | 26 Mar 2019 11:21 |
Published Version: | https://arxiv.org/abs/1809.10628v2 |
Status: | Submitted |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:143439 |