Determinantal Barlow surfaces and phantom categories

Determinantal Barlow surfaces and phantom categories I will report on the construction of an exceptional collection of maximal length 11 on the Barlow surface S. This can be used to 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 and a category with trivial Grothendieck group and Hochschild homology, called a phantom category. This is joint work with Christian B?hning, Hans-Christian Graf von Bothmer and Ludmil Katzarkov.