On nodal Enriques surfaces and quartic double solids

TL;DR

This paper explores the relationship between certain singular double covers of projective space, known as quartic double solids, and associated nodal Enriques surfaces, revealing a derived category equivalence that links their geometric structures.

Contribution

It establishes a derived category equivalence between the nontrivial parts of Enriques surfaces and minimal resolutions of specific quartic double solids, connecting their geometric and categorical properties.

Findings

Derived category of Enriques surface is equivalent to that of the resolution of the quartic double solid.

Identifies a categorical link between nodal Enriques surfaces and quartic double solids.

Provides new insights into the geometric structures of unirational but nonrational conic bundles.

Abstract

We consider the class of singular double coverings $X \to \PP^3$ 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 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 $X$.