Exceptional collections on $\Sigma _{ 2 }$

Structure theorems for exceptional objects and exceptional collections of the bounded derived category of coherent sheaves on del Pezzo surfaces are established by Kuleshov and Orlov. In this paper we propose conjectures which generalize these results to weak del Pezzo surfaces. Unlike del Pezzo surfaces, an exceptional object on a weak del Pezzo surface is not necessarily a shift of a sheaf and is not determined by its class in the Grothendieck group. Our conjectures explain how these complications are taken care of by spherical twists, the categorification of $(-2)$-reflections acting on the derived category. This paper is devoted to solving the conjectures for the prototypical weak del Pezzo surface $\Sigma _{ 2 }$, the Hirzebruch surface of degree $2$. Specifically, we prove the following results: Any exceptional object is sent to the shift of the uniquely determined exceptional vector bundle by a product of spherical twists which acts trivially on the Grothendieck group of the derived category. Any exceptional collection on $\Sigma _{ 2 }$ is part of a full exceptional collection. We moreover prove that the braid group on $4$ strands acts transitively on the set of exceptional collections of length $4$ (up to shifts).