Determinantal Barlow surfaces and phantom categories

  • Christian Böhning

    Universität Hamburg, Germany
  • Hans-Christian Graf von Bothmer

    Universität Hamburg, Germany
  • Ludmil Katzarkov

    University of Miami, Coral Gables, USA
  • Pawel Sosna

    Universität Hamburg, Germany


We prove that the bounded derived category of the surface constructed by Barlow admits a length 11 exceptional sequence consisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of 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 itself is not full and its (left or right) orthogonal complement is also a phantom category.

Cite this article

Christian Böhning, Hans-Christian Graf von Bothmer, Ludmil Katzarkov, Pawel Sosna, Determinantal Barlow surfaces and phantom categories. J. Eur. Math. Soc. 17 (2015), no. 7, pp. 1569–1592

DOI 10.4171/JEMS/539