Determinantal Barlow surfaces and phantom categories
Christian Böhning
Universität Hamburg, GermanyHans-Christian Graf von Bothmer
Universität Hamburg, GermanyLudmil Katzarkov
University of Miami, Coral Gables, USAPawel Sosna
Universität Hamburg, Germany
Abstract
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