Exceptional sequences of maximal length on some surfaces isogenous to a higher product

2015

Journal of Algebra

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Section: Discussionmentioning

confidence: 99%

Derived categories of surfaces isogenous to a higher product

2015

Journal of Algebra

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“…From the above computations we also have the following Lemma. bundles on D of degree 8(see [13], [14], [23] for more details). We also consider Proof.…”

Section: Constructing Line Bundles On Dmentioning

confidence: 99%

Quasiphantom categories on a family of surfaces isogenous to a higher product

Kim

2017

Journal of Algebra

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We construct exceptional collections of line bundles of maximal length 4 on S = (C ×D)/G which is a surface isogenous to a higher product with p g = q = 0 where G = G(32, 27) is a finite group of order 32 having number 27 in the list of Magma library. From these exceptional collections, we obtain new examples of quasiphantom categories as their orthogonal complements.A surface S which is isomorphic to (C ×D)/G where C, D are curves of genus ≥ 2 and G is a finite group acting on C × D freely is called a surface isogenous to a higher product. Surfaces isogenous to a higher product are interesting and important classes of surfaces of general type. They play an important role in the study of moduli spaces of general type surfaces. Bauer, Catanese and Grunewald classified surfaces isogenous to a higher product of unmixed type with p g = q = 0 in [4]. In particular, they proved that the possible list ofA 5 cases. It is very natural to expect that derived categories of all surfaces isogenous to a higher product with p g = q = 0 have quasiphantom categories in their derived categories. However quasiphantom categories in derived categories of surfaces had not been constructed for G = G(32, 27), A 5 cases.In this paper we construct exceptional collections of line bundles of maximal length 4 on S = (C × D)/G which is a surface isogenous to a higher product with p g = q = 0 and G is G(32, 27). From these exceptional collections we can obtain new examples of quasiphantom categories. They are obtained as the orthogonal complements of the exceptional collections of maximal length 4 on S.Acknowledgement. We are grateful to Changho Keem, Young-Hoon Kiem for their words of encouragement. We thank Fabrizio Catanese for answering many questions and helpful discussions. We also thank Alexander Kuznetsov for his suggestion to use a computer program to compute the derived categories of the family of surfaces considered in this paper. Part of this work was done when the third named author was a research fellow of Korea Institute for Advanced Study. He thanks KIAS for wonderful working condition and kind hospitality.This work was also supported by IBS-R003-Y1. Last but not least, we thank the referees for their valuable comments which helped us to improve the manuscript.Notations. We will work over C. Derived category of a variety will mean the

“…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.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore the maximal possible length of the exceptional sequence on every such surface is 4. For the 4 families of such surfaces with abelian group quotients, exceptional collections of maximal length were constructed in [12,15,16]. In this paper we construct such collections in four more cases where G is D 4 ×Z/2, S 4 , S 4 ×Z/2 and (Z/4×Z/2)⋊Z/2 (G(16) in the notation of [6]).…”

Section: Introductionmentioning

confidence: 99%

Exceptional collections on some fake quadrics

Shabalin

2018

Proc. Amer. Math. Soc.

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We construct exceptional collections of maximal length on four families of surfaces of general type with pg = q = 0 which are isogenous to a product of curves. From these constructions we obtain new examples of quasiphantom categories as their orthogonal complements.Question. Let S be a smooth projective surface of general type with p g = q = 0. Is there an exceptional sequence whose length is equal to the rank of Grothendieck Key words and phrases. Derived category, exceptional sequence, quasiphantom category, surfaces of general type, surfaces isogenous to a higher product.