Exceptional collections of line bundles on the Beauville surface

TL;DR

This paper constructs special subcategories in the derived category of the Beauville surface, identifying all maximal-length line bundle exceptional collections and revealing their structure as helices.

Contribution

It explicitly classifies all maximal-length line bundle exceptional collections on the Beauville surface and describes their arrangement as helices.

Findings

Six such exceptional collections exist (up to twist)

These collections form two helices

Quasi-phantom subcategories are constructed as orthogonals

Abstract

We construct quasi-phantom admissible subcategories in the derived category of coherent sheaves on the Beauville surface $S$. These quasi-phantoms subcategories appear as right orthogonals to subcategories generated by exceptional collections of maximal possible length 4 on $S$. We prove that there are exactly 6 exceptional collections consisting of line bundles (up to a twist) and these collections are spires of two helices.