### Abstract

We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov's results on heights of exceptional sequences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category.

Original language | English (US) |
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Pages (from-to) | 1569-1592 |

Number of pages | 24 |

Journal | Journal of the European Mathematical Society |

Volume | 17 |

Issue number | 7 |

DOIs | |

State | Published - 2015 |

### Fingerprint

### Keywords

- Barlow surfaces
- Derived categories
- Exceptional collections
- Hochschild homology
- Semiorthogonal decompositions

### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Journal of the European Mathematical Society*,

*17*(7), 1569-1592. https://doi.org/10.4171/JEMS/539

**Determinantal Barlow surfaces and phantom categories.** / Böhning, Christian; Graf Von Bothmer, Hans Christian; Katzarkov, Ludmil; Sosna, Pawel.

Research output: Contribution to journal › Article

*Journal of the European Mathematical Society*, vol. 17, no. 7, pp. 1569-1592. https://doi.org/10.4171/JEMS/539

}

TY - JOUR

T1 - Determinantal Barlow surfaces and phantom categories

AU - Böhning, Christian

AU - Graf Von Bothmer, Hans Christian

AU - Katzarkov, Ludmil

AU - Sosna, Pawel

PY - 2015

Y1 - 2015

N2 - We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov's results on heights of exceptional sequences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category.

AB - We prove that the bounded derived category of the surface S constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow surfaces, the generic surface has a semiorthogonal decomposition of its derived category into a length 11 exceptional sequence of line bundles and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is done using a deformation argument and the fact that the derived endomorphism algebra of the sequence is constant. Applying Kuznetsov's results on heights of exceptional sequences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category.

KW - Barlow surfaces

KW - Derived categories

KW - Exceptional collections

KW - Hochschild homology

KW - Semiorthogonal decompositions

UR - http://www.scopus.com/inward/record.url?scp=84932164956&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84932164956&partnerID=8YFLogxK

U2 - 10.4171/JEMS/539

DO - 10.4171/JEMS/539

M3 - Article

AN - SCOPUS:84932164956

VL - 17

SP - 1569

EP - 1592

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

SN - 1435-9855

IS - 7

ER -