Enumerating exceptional collections of line bundles on some surfaces of general type

Abstract

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 2 X = 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.

Derived categories of surfaces isogenous to a higher product

“…We also compute Hochschild cohomologies of quasiphantom categories and prove that for some exceptional sequences we obtained the categories generated by those exceptional sequences are defomation invariant. While adding these results to this paper which was on the arXiv, similar results have been obtained independently by Coughlan in [10] via different method. In his paper [10], Coughlan considers general type surfaces which are obtained as abelian covers of del Pezzo surfaces satisfying some conditions.…”

“…While adding these results to this paper which was on the arXiv, similar results have been obtained independently by Coughlan in [10] via different method. In his paper [10], Coughlan considers general type surfaces which are obtained as abelian covers of del Pezzo surfaces satisfying some conditions. His method can be applied to surfaces isogenous to a higher product with G = (Z/3) 2 , G = (Z/5) 2 and many other general type surfaces.…”

“…They constructed exceptional sequences of maximal length for primary Burniat surfaces and proved that the above property holds for their exceptional sequences in [1]. This phenomenon was observed for other general type surfaces ( see [8], [9], [10] ). We prove that for certain exceptional sequence we constructed the above question is true.…”

“…Recently derived categories of surfaces of general type draw lots of attention. Several interesting semiorthogonal decompositions of the derived categories were constructed by Böhning, Graf von Bothmer, and Sosna on the classical Godeaux surface [9]; Alexeev and Orlov on the primary Burniat surfaces [1]; Galkin and Shinder on the Beauville surface [15]; Böhning, Graf von Bothmer, Katzarkov and Sosna on the determinantal Barlow surfaces [8]; Fakhruddin on some fake projective planes [14]; Galkin, Katzarkov, Mellit and Shinder on some different fake projective planes and on a fake cubic surface [16]; Coughlan on some surfaces obtained as abelian coverings of del Pezzo surfaces [10]. These semiorthogonal decompositions consist of admissible subcategories generated by exceptional sequences of line bundles of maximal lengths and their orthogonal complements.…”

Exceptional collections on some fake quadrics

“…Motivated by their results now there are lots of studies on derived categories of surfaces of general type with p g = q = 0. See the papers of Böhning, Graf von Bothmer, and Sosna [4], Alexeev and Orlov [1], Galkin and Shinder [12], Böhning, Graf von Bothmer, Katzarkov and Sosna [3], Fakhruddin [10], Galkin, Katzarkov, Mellit and Shinder [11], Coughlan [8], Keum [14] and the first author [15,16] for more details. They constructed categories with vanishing Hochschild homologies as orthogonal complements of exceptional sequences of line bundles of maximal lengths.…”