Determinantal Barlow surfaces and phantom categories

Christian Böhning, Hans Christian Graf Von Bothmer, Ludmil Katzarkov, Pawel Sosna

Research output: Contribution to journalArticle

12 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)1569-1592
Number of pages24
JournalJournal of the European Mathematical Society
Volume17
Issue number7
DOIs
StatePublished - 2015

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Phantom
Derived Category
Line Bundle
Algebra
Grothendieck Group
Hochschild Homology
Decomposition
Endomorphism
Moduli Space
Trivial
Complement
Decompose

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

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

In: Journal of the European Mathematical Society, Vol. 17, No. 7, 2015, p. 1569-1592.

Research output: Contribution to journalArticle

Böhning, Christian ; Graf Von Bothmer, Hans Christian ; Katzarkov, Ludmil ; Sosna, Pawel. / Determinantal Barlow surfaces and phantom categories. In: Journal of the European Mathematical Society. 2015 ; Vol. 17, No. 7. pp. 1569-1592.
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