Enumerating Exceptional Collections of Line Bundles on Some Surfaces of General Type

We use constructions of surfaces as abelian covers to write down exceptional
collections of line bundles of maximal length for every surface $X$ in
certain families of surfaces of general type with $p_{g}=0$ and $K_{X}^{2}=3,4,5,6,8$.
We also compute the algebra of derived endomorphisms for an appropriately
chosen exceptional collection, and the Hochschild cohomology of the corresponding
quasiphantom category. As a consequence, we see that the subcategory generated
by the exceptional collection does not vary in the family of surfaces.
Finally, we describe the semigroup of effective divisors on each surface,
answering a question of Alexeev.

2010 Mathematics Subject Classification: 14F05 (14J29)

Keywords and Phrases: Derived category; Kulikov surface; Burniat surface; Beauville surface; Semiorthogonal decomposition; Exceptional sequence; Hochschild homology

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