Exceptional collections of line bundles on the Beauville surface

Abstract: It is well known that the derived category of coherent sheaves on a quadric surface has a full exceptional collection of 4 line bundles.

We study the derived category of the Beauville surface $S$, which is a surface of general type having the same numerical invariants as the quadric. We construct (non-full) 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.

 

Right orthogonals to our exceptional collections are quasi-phantom admissible subcategories in the derived category of $S$: They have vanishing Hochschild homology and finite abelian K_0. This is a joint work with Sergey Galkin.