Determinantal Barlow surfaces and phantom categories

TL;DR

This paper investigates the derived categories of determinantal Barlow surfaces, establishing the existence of exceptional sequences and phantom categories, and analyzing their deformation behavior and semiorthogonal decompositions.

Contribution

It proves the existence of explicit length 11 exceptional sequences on Barlow surfaces and shows that nearby surfaces have semiorthogonal decompositions involving phantom categories.

Findings

Existence of explicit length 11 exceptional sequences on Barlow surfaces

Nearby surfaces have semiorthogonal decompositions with phantom categories

The exceptional sequence on S is not full and has a phantom orthogonal complement

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.