Derived categories of cubic and V14 threefolds

TL;DR

This paper establishes a deep categorical equivalence between certain derived categories of V14 Fano threefolds and smooth cubic threefolds, revealing a new geometric correspondence involving vector bundles and flops.

Contribution

It introduces a novel geometric correspondence between V14 threefolds and cubic threefolds via vector bundles and flops, proving derived category equivalences.

Findings

Derived categories of V14 threefolds and cubic threefolds are equivalent.

A natural flop relates vector bundles on V14 and cubic threefolds.

The correspondence allows reconstruction of V14 threefolds from cubic 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.