Abstract
We prove that two Enriques surfaces defined over an algebraically closed field of characteristic different from 2 are isomorphic if their Kuznetsov components are equivalent. This improves and completes our previous result joint with Nuer where the same statement is proved for generic Enriques surfaces.
Highlights
The bounded derived category of coherent sheaves Db(X ) of an Enriques surface X has been widely investigated
A very nice result by Bridgeland and Maciocia [5] shows that the derived category determines the surface up to isomorphism. This goes under the name of Derived Torelli Theorem as it can be viewed as a categorification of the usual Hodge-theoretic Torelli Theorem for Enriques surfaces
From the geometric point of view, they are quotients of a K3 surface by a fixed-point-free involution. It is a fact [13] that the Derived Torelli Theorem above holds in any characteristic different from 2 as soon as the field is algebraically closed
Summary
The bounded derived category of coherent sheaves Db(X ) of an Enriques surface X has been widely investigated. From the geometric point of view, they are quotients of a K3 surface by a fixed-point-free involution It is a fact [13] that the Derived Torelli Theorem above holds in any characteristic different from 2 as soon as the field is algebraically closed. The strategy was to deform two Enriques surface X1 and X2 containing (−2)-curves and with an equivalence of Fourier–Mukai type Ku(X1, L1) ∼= Ku(X2, L) to generic unnodal Enriques surfaces and apply the generic Refined Derived Torelli Theorem in [22] Even though this strategy is still valid, a precise implementation would require a careful (and possibly complicated) comparison between the deformation theory of the Enriques surface and of its Kuznetsov component. We feel like that the approach in the present paper is more direct and conceptually clearer
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