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Exceptional collections on some fake quadrics

Authors: Kyoung-Seog Lee and Timofey Shabalin
Journal: Proc. Amer. Math. Soc. 146 (2018), 2299-2313
MSC (2010): Primary 14F05; Secondary 14J29
Published electronically: March 9, 2018
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Abstract: We construct exceptional collections of maximal length on four families of surfaces of general type with $ p_g=q=0$ which are isogenous to a product of curves. From these constructions we obtain new examples of quasiphantom categories as their orthogonal complements.

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  • [1] Valery Alexeev and Dmitri Orlov, Derived categories of Burniat surfaces and exceptional collections, Math. Ann. 357 (2013), no. 2, 743-759. MR 3096524,
  • [2] Arnaud Beauville, Vanishing thetanulls on curves with involutions, Rend. Circ. Mat. Palermo (2) 62 (2013), no. 1, 61-66. MR 3031568,
  • [3] Christian Böhning, Hans-Christian Graf von Bothmer, Ludmil Katzarkov, and Pawel Sosna, Determinantal Barlow surfaces and phantom categories, J. Eur. Math. Soc. (JEMS) 17 (2015), no. 7, 1569-1592. MR 3361723,
  • [4] Christian Böhning, Hans-Christian Graf von Bothmer, and Pawel Sosna, On the derived category of the classical Godeaux surface, Adv. Math. 243 (2013), 203-231. MR 3062745,
  • [5] Ingrid C. Bauer and Fabrizio Catanese, Some new surfaces with $ p_g=q=0$, The Fano Conference, Univ. Torino, Turin, 2004, pp. 123-142. MR 2112572
  • [6] I. C. Bauer, F. Catanese, and F. Grunewald, The classification of surfaces with $ p_g=q=0$ isogenous to a product of curves, Pure Appl. Math. Q. 4 (2008), no. 2, Special Issue: In honor of Fedor Bogomolov, 547-586. MR 2400886,
  • [7] Fabrizio Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000), no. 1, 1-44. MR 1737256
  • [8] Stephen Coughlan, Enumerating exceptional collections of line bundles on some surfaces of general type, Doc. Math. 20 (2015), 1255-1291. MR 3424480
  • [9] Igor V. Dolgachev, Invariant stable bundles over modular curves $ X(p)$, Recent progress in algebra (Taejon/Seoul, 1997) Contemp. Math., vol. 224, Amer. Math. Soc., Providence, RI, 1999, pp. 65-99. MR 1653063,
  • [10] Najmuddin Fakhruddin, Exceptional collections on 2-adically uniformized fake projective planes, Math. Res. Lett. 22 (2015), no. 1, 43-57. MR 3342178,
  • [11] Sergey Galkin, Ludmil Katzarkov, Anton Mellit, and Evgeny Shinder, Derived categories of Keum's fake projective planes, Adv. Math. 278 (2015), 238-253. MR 3341791,
  • [12] Sergey Galkin and Evgeny Shinder, Exceptional collections of line bundles on the Beauville surface, Adv. Math. 244 (2013), 1033-1050. MR 3077896,
  • [13] Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR 0463157
  • [14] J. Keum, A vanishing theorem on fake projective planes with enough automorphisms, arXiv:1407.7632.
  • [15] Kyoung-Seog Lee, Derived categories of surfaces isogenous to a higher product, J. Algebra 441 (2015), 180-195. MR 3391925,
  • [16] Kyoung-Seog Lee, Exceptional sequences of maximal length on some surfaces isogenous to a higher product, J. Algebra 454 (2016), 308-333. MR 3473430,
  • [17] Gregory Karpilovsky, The Schur multiplier, London Mathematical Society Monographs. New Series, vol. 2, The Clarendon Press, Oxford University Press, New York, 1987. MR 1200015
  • [18] A. Kuznetsov, Hochschild homology and semiorthogonal decompositions, preprint, 2009, arXiv:0904.4330.
  • [19] Alexander Kuznetsov, Height of exceptional collections and Hochschild cohomology of quasiphantom categories, J. Reine Angew. Math. 708 (2015), 213-243. MR 3420334,
  • [20] Shinnosuke Okawa, Semi-orthogonal decomposability of the derived category of a curve, Adv. Math. 228 (2011), no. 5, 2869-2873. MR 2838062,
  • [21] Rita Pardini, The classification of double planes of general type with $ K^2=8$ and $ p_g=0$, J. Algebra 259 (2003), no. 1, 95-118. MR 1953710,
  • [22] Charles A. Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, vol. 38, Cambridge University Press, Cambridge, 1994. MR 1269324

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Additional Information

Kyoung-Seog Lee
Affiliation: Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea

Timofey Shabalin
Affiliation: National Research University Higher School of Economics, AG Laboratory, 7 Vavilova street, Moscow, Russia, 117312

Keywords: Derived category, exceptional sequence, quasiphantom category, surfaces of general type, surfaces isogenous to a higher product
Received by editor(s): January 22, 2016
Received by editor(s) in revised form: October 5, 2016
Published electronically: March 9, 2018
Additional Notes: The first author was supported by Seoul National University via the Fellowship for Fundamental Academic Fields. He was supported by IBS-R003-Y1
The second author was partially supported by AG Laboratory HSE, RF government grant, ag. 11.G34.31.0023 and RScF grant, ag. 14-21-00053
Communicated by: Lev Borisov
Article copyright: © Copyright 2018 American Mathematical Society

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