Mathematische Annalen

, Volume 361, Issue 1–2, pp 107–133 | Cite as

On nodal Enriques surfaces and quartic double solids

  • Colin Ingalls
  • Alexander KuznetsovEmail author


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 \(X\).



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.


  1. 1.
    Artin, M., Mumford, D.: Some elementary examples of unirational varieties which are not rational. Proc. Lond. Math. Soc. 3, 75–95 (1972)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Atiyah, M.: On analytic surfaces with double points. Proc. R. Soc. Lond. Ser. A 247, 237–244 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Beauville, A.: Complex algebraic surfaces, LMSST, 2nd edn. Cambridge University Press, Cambridge (1996)CrossRefGoogle Scholar
  4. 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)Google Scholar
  5. 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)Google Scholar
  6. 6.
    Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties, preprint math.AG/9506012Google Scholar
  7. 7.
    Bondal, A., Orlov, D.: Reconstruction of a variety from the derived category and groups of autoequivalences. Compos. Math. 125(3), 327–344 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Cossec, F.: Reye congruences. Trans. AMS 280(2), 737–751 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Cossec, F., Dolgachev, I.: Enriques Surfaces I, Progress in Mathematics, 76. Birkhäuser Boston, Inc., Boston (1989)CrossRefGoogle Scholar
  10. 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)Google Scholar
  11. 11.
    Fulton, W., Harris, J.: Representation Theory: A First Course, vol. 129. Springer, Berlin (1991)zbMATHGoogle Scholar
  12. 12.
    Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math. 218(5), 1340–1369 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Kuznetsov, A.: Calabi-Yau categories, unpublishedGoogle Scholar
  14. 14.
    Kuznetsov, A.: Scheme of lines on a family of quadrics: geometry and derived category. Math. Zeitschrift 276(3), 655–672 (2014)CrossRefzbMATHGoogle Scholar
  15. 15.
    Moishezon, B.: Algebraic varieties and compact complex spaces. In: Proceedings International Congress Mathematicians (Nice, 1970) 2, Gauthier-Villars, pp. 643–648Google Scholar
  16. 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)Google Scholar
  17. 17.
    Zube, S.: Exceptional vector bundles on Enriques surfaces. Math. Notes 61(6), 693–699 (1997)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of New BrunswickFrederictonCanada
  2. 2.Algebraic Geometry SectionSteklov Mathematical InstituteMoscowRussia
  3. 3.The Poncelet LaboratoryIndependent University of MoscowMoscowRussia
  4. 4.Laboratory of Algebraic GeometrySU-HSEMoscowRussia

Personalised recommendations