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On nodal Enriques surfaces and quartic double solids

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Abstract

We consider the class of singular double coverings \(X \rightarrow {\mathbb {P}}^3\) ramified in the degeneration locus \(D\) of a family of 2-dimensional quadrics. These are precisely the quartic double solids constructed by Artin and Mumford as examples of unirational but nonrational conic bundles. With such a quartic surface \(D,\) one can associate an Enriques surface \(S\) which is the factor of the blowup of \(D\) by a natural involution acting without fixed points (such Enriques surfaces are known as nodal Enriques surfaces or Reye congruences). We show that the nontrivial part of the derived category of coherent sheaves on this Enriques surface \(S\) is equivalent to the nontrivial part of the derived category of a minimal resolution of singularities of \(X\).

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References

  1. Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 3, 75–95 (1972)

    Article  MathSciNet  Google Scholar 

  2. Atiyah, M.: On analytic surfaces with double points. Proc. R. Soc. Lond. Ser. A 247, 237–244 (1958)

    Article  MATH  MathSciNet  Google Scholar 

  3. Beauville, A.: Complex algebraic surfaces, LMSST, 2nd edn. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  4. Bondal, A.: Representations of associative algebras and coherent sheaves, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. vol. 53, no. 1, 25–44 (1989); translation in Math. USSR-Izv. vol. 34, no. 1, pp. 23–42 (1990)

  5. Bondal, A., Kapranov, M.: Representable functors, Serre functors, and reconstructions, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53, no. 6, pp. 1183–1205, 1337 (1989); translation in Math. USSR-Izv. Vol. 35, no. 3, pp. 519–541 (1990)

  6. Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties, preprint math.AG/9506012

  7. Bondal, A., Orlov, D.: Reconstruction of a variety from the derived category and groups of autoequivalences. Compos. Math. 125(3), 327–344 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cossec, F.: Reye congruences. Trans. AMS 280(2), 737–751 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cossec, F., Dolgachev, I.: Enriques Surfaces I, Progress in Mathematics, 76. Birkhäuser Boston, Inc., Boston (1989)

    Book  Google Scholar 

  10. Dolgachev, I., Reider, I.: On rank 2 vector bundles with \(c_1^2=10\) and \(c_2=3\) on Enriques surfaces, Algebraic Geometry (Chicago, IL, 1989) 39–49, Lectures Notes in Math. 1479, Springer, Berlin (1991)

  11. Fulton, W., Harris, J.: Representation Theory: A First Course, vol. 129. Springer, Berlin (1991)

    MATH  Google Scholar 

  12. Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218(5), 1340–1369 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Kuznetsov, A.: Calabi-Yau categories, unpublished

  14. Kuznetsov, A.: Scheme of lines on a family of quadrics: geometry and derived category. Math. Zeitschrift 276(3), 655–672 (2014)

    Article  MATH  Google Scholar 

  15. Moishezon, B.: Algebraic varieties and compact complex spaces. In: Proceedings International Congress Mathematicians (Nice, 1970) 2, Gauthier-Villars, pp. 643–648

  16. Orlov, D.: Projective bundles, monoidal transformations, and derived categories of coherent sheaves, (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 56(4) 852–862 (1992); translation in. Russian Acad. Sci. Izv. Math. 41(1), pp. 133–141 (1993)

  17. Zube, S.: Exceptional vector bundles on Enriques surfaces. Math. Notes 61(6), 693–699 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

The first author would like to thank A. Bondal for helpful discussions. The second author would like to thank L. Katzarkov and D. Orlov for helpful discussions and is very grateful to I. Dolgachev for sharing many interesting facts about Enriques surfaces. We would also like to thank the referee for many helpful comments on an earlier version of this paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of New Brunswick, Fredericton, NB, E3B 5A3, Canada

    Colin Ingalls

  2. Algebraic Geometry Section, Steklov Mathematical Institute, 8 Gubkin str., Moscow,  119991, Russia

    Alexander Kuznetsov

  3. The Poncelet Laboratory, Independent University of Moscow, Moscow, Russia

    Alexander Kuznetsov

  4. Laboratory of Algebraic Geometry, SU-HSE, Moscow, Russia

    Alexander Kuznetsov

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  1. Colin Ingalls
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  2. Alexander Kuznetsov
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Correspondence to Alexander Kuznetsov.

Additional information

C.I. was partially supported by a NSERC Discovery Grant.

A.K. was partially supported by RFFI grant NSh-2998.2014.1 and by AG Laboratory SU-HSE, RF government grant, ag.11.G34.31.0023.

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Ingalls, C., Kuznetsov, A. On nodal Enriques surfaces and quartic double solids. Math. Ann. 361, 107–133 (2015). https://doi.org/10.1007/s00208-014-1066-y

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  • DOI: https://doi.org/10.1007/s00208-014-1066-y

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