# 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 \(X\).

## Notes

### 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.

## References

- 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.Atiyah, M.: On analytic surfaces with double points. Proc. R. Soc. Lond. Ser. A
**247**, 237–244 (1958)CrossRefzbMATHMathSciNetGoogle Scholar - 3.Beauville, A.: Complex algebraic surfaces, LMSST, 2nd edn. Cambridge University Press, Cambridge (1996)CrossRefGoogle 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)Google Scholar
- 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.Bondal, A., Orlov, D.: Semiorthogonal decomposition for algebraic varieties, preprint math.AG/9506012Google Scholar
- 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.Cossec, F.: Reye congruences. Trans. AMS
**280**(2), 737–751 (1983)CrossRefzbMATHMathSciNetGoogle Scholar - 9.Cossec, F., Dolgachev, I.: Enriques Surfaces I, Progress in Mathematics, 76. Birkhäuser Boston, Inc., Boston (1989)CrossRefGoogle 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)Google Scholar
- 11.Fulton, W., Harris, J.: Representation Theory: A First Course, vol. 129. Springer, Berlin (1991)zbMATHGoogle Scholar
- 12.Kuznetsov, A.: Derived categories of quadric fibrations and intersections of quadrics. Adv. Math.
**218**(5), 1340–1369 (2008)CrossRefzbMATHMathSciNetGoogle Scholar - 13.Kuznetsov, A.: Calabi-Yau categories, unpublishedGoogle Scholar
- 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.Moishezon, B.: Algebraic varieties and compact complex spaces. In: Proceedings International Congress Mathematicians (Nice, 1970) 2, Gauthier-Villars, pp. 643–648Google Scholar
- 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.Zube, S.: Exceptional vector bundles on Enriques surfaces. Math. Notes
**61**(6), 693–699 (1997)CrossRefzbMATHMathSciNetGoogle Scholar