# 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.

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