Derived categories of cubic and V14 threefolds

Abstract

We show that the projectivization of the exceptional rank 2 vector bundle on an arbitrary smooth V14 Fano threefold after a certain natural flop turns into the projectivization of an instanton vector bundle on a smooth cubic threefold. And vice versa, starting from a smooth cubic threefold with an instanton vector bundle of charge 2 on it we reconstruct V14 threefold. Relying on the geometric properties of the above correspondence we prove that the orthogonals to the exceptional pairs in the bounded derived categories of coherent sheaves on a smooth V14 threefold and on the corresponding cubic threefold are equivalent as triangulated categories.

24 pages, dedicated to the memory of Andrei Nikolaevich Tyurin

Cited by in corpus (20)

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• Stability conditions on Kuznetsov components
• The Fano variety of lines and rationality problem for a cubic hypersurface
• Categories of Massless D-Branes and del Pezzo Surfaces
• Fano manifolds of Calabi-Yau type
• Some remarks on Fano threefolds of index two and stability conditions
• Compactification of the moduli space of instanton sheaves on the Fano threefold $V_5$
• Hochschild cohomology versus the Jacobian ring, and the Torelli theorem for cubic fourfolds
• Bridgeland Moduli spaces for Gushel-Mukai threefolds and Kuznetsov's Fano threefold conjecture
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• Lagrangian embeddings of cubic fourfolds containing a plane
• Derived categories and rationality of conic bundles
• Voevodsky's conjecture for cubic fourfolds and Gushel-Mukai fourfolds via noncommutative K3 surfaces
• Frobenius and spherical codomains and neighbourhoods
• Derived autoequivalences and a weighted Beilinson resolution