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 con-sisting of (explicit) line bundles. Moreover, we show that in a small neighbourhood of S in the moduli space of determinantal Barlow sur-faces, the generic surface has a semiorthogonal decomposition of its de-rived 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 se-quences, we also show that the sequence on S itself is not full and its (left or right) orthogonal complement is also a phantom category. 1