Fourier-Mukai partners for very general special cubic fourfolds

TL;DR

This paper constructs explicit examples of very general special cubic fourfolds with associated K3 surfaces that have multiple non-isomorphic Fourier-Mukai partners, revealing new relationships between their derived categories.

Contribution

It provides explicit examples and formulas for the number of Fourier-Mukai partners of special cubic fourfolds based on their discriminant and associated K3 surfaces.

Findings

Number of Fourier-Mukai partners equals that of the associated K3 surface when discriminant d ≡ 2 mod 6.

When discriminant d ≡ 0 mod 6, the cubic fourfold has roughly half as many Fourier-Mukai partners.

Explicit examples demonstrate the existence of non-isomorphic Fourier-Mukai partners for very general special cubic fourfolds.

Abstract

We exhibit explicit examples of very general special cubic fourfolds with discriminant $d$ admitting an associated (twisted) K3 surface, which have non-isomorphic Fourier-Mukai partners. In particular, in the untwisted setting, we show that the number of Fourier-Mukai partners for a very general special cubic fourfold with discriminant $d$ and having an associated K3 surface, is equal to the number $m$ of Fourier-Mukai partners of its associated K3 surface, if $d \equiv 2 (\text{mod}\,6)$; else, if $d \equiv 0 (\text{mod}\,6)$, the cubic fourfold has $\lceil m/2 \rceil$ Fourier-Mukai partners.