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

TL;DR

This work addresses the problem of understanding birational and categorical relations among smooth cubic fourfolds by focusing on non-syzygetic cubics and making the Fourier-Mukai partner construction explicit via Gale duality. The authors prove that a very general non-syzygetic cubic fourfold $X$ has exactly one nontrivial Fourier-Mukai partner $X'$, which is obtained as the Gale dual, and they connect this pair to Gushel-Mukai fourfolds and EPW sextics through a robust linear-algebra/Lagrangian framework. They show that Gale dual cubics are birational and share birational Fano varieties of lines under generic conditions, and they extend these phenomena to an equivariant setting, highlighting both compatibility and potential counterexamples with conjectures in $G$-birational geometry. The paper also provides a representation-theoretic reformulation and gives explicit $G$-examples (notably with $A_4$ actions) to probe equivariant forms of the conjectures, offering concrete computable criteria and linking the construction to prior work of Brooke–Frei–Marquand and Kuznetsov–Perry. Overall, the results yield a concrete, computable pathway to verify FM-partner phenomena and birationality while clarifying the role of symmetries in these equivalences.

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.

Theorems & Definitions (47)

• Theorem 1.1
• Definition 2.1
• Proposition 2.2
• proof
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4