Abstract
We prove that two general Enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their Kuznetsov components are equivalent. We apply the same techniques to give a new simple proof of a conjecture by Ingalls and Kuznetsov relating the derived categories of the blow-up of general Artin–Mumford quartic double solids and of the associated Enriques surfaces.
Highlights
An Enriques surface is a smooth projective surface X with 2-torsion dualizing sheaf ωX and such that H 1(X, OX ) = 0
In characteristic zero, Hodgetheoretic Torelli theorems hold for both Enriques and K3 surfaces
If we increase the dimension of the hypersurfaces by one and we consider two cubic fourfolds W1 and W2, we get semiorthogonal decompositions similar to the one above but with additional exceptional objects OWi (2Hi )
Summary
We assume that all varieties are defined over an algebraically closed field K of characteristic different from 2 Under this additional assumption, the above definition is equivalent to asking that X is the quotient of a K3 surface by a fixed-point-free involution. If we increase the dimension of the hypersurfaces by one and we consider two cubic fourfolds W1 and W2, we get semiorthogonal decompositions similar to the one above but with additional exceptional objects OWi (2Hi ). In this case, the admissible subcategories Ku(Wi ) behave like the derived category of a K3 surface. Theorem A is further evidence of the fact that, when these decompositions originate from geometry, they usually encode important pieces of information
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