Determinantal Barlow surfaces and phantom categories
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.
Cite
@article{arxiv.1210.0343,
title = {Determinantal Barlow surfaces and phantom categories},
author = {Christian Böhning and Hans-Christian Graf von Bothmer and Ludmil Katzarkov and Pawel Sosna},
journal= {arXiv preprint arXiv:1210.0343},
year = {2017}
}
Comments
27 pages; 1 figure; Macaulay2 code for the paper available at http://www.math.uni-hamburg.de/home/boehning/research/BarlowM2/M2scripts