Fourier-Mukai partners of non-syzygetic cubic fourfolds and Gale duality

TL;DR

This paper investigates non-syzygetic cubic fourfolds, showing they have unique Fourier-Mukai partners related via Gale duality, and explores their birational properties and symmetries, providing new insights into their geometric and algebraic structures.

Contribution

It characterizes non-syzygetic cubic fourfolds algebraically, links their Fourier-Mukai partners through Gale duality, and examines their birational relationships and symmetry actions.

Findings

Non-syzygetic cubic fourfolds have exactly one nontrivial Fourier-Mukai partner.

Gale duality relates the equations of dual cubic fourfolds.

Gale dual cubics are birational and have birational Fano varieties of lines under generic conditions.

Abstract

We study so-called non-syzygetic cubic fourfolds, i.e., smooth cubic fourfolds containing two cubic surface scrolls in distinct hyperplanes with intersection number between the two scrolls equal to $1$. We prove that a very general non-syzygetic cubic fourfold has precisely one nontrivial Fourier-Mukai partner that is also non-syzygetic. We characterise non-syzygetic cubic fourfolds algebraically as those having a special type of equation that is almost linear determinantal, and show that the equation of the Fourier-Mukai partner can be obtained by applying Gale duality. We establish that Gale dual cubics are birational, Fourier-Mukai partners and have birational Fano varieties of lines under suitable genericity assumptions, recovering a result of Brooke-Frei-Marquand. We show that the birationality of the Fano varieties of lines continues to hold in the context of equivariant birational geometry, but birationality of the cubics may not. We exhibit examples of Gale dual cubics with faithful actions of the alternating group on four letters that could provide counterexamples to equivariant versions of a conjecture by Brooke-Frei-Marquand predicting birationality of the cubics if the Fano varieties of lines are birational, and also possibly a related conjecture by Huybrechts predicting birationality of Fourier-Mukai partners.