Distinguish birational classes of irrational cubic fourfolds

Determine how to distinguish the birational class of one irrational cubic fourfold from that of another, particularly for non-isomorphic irrational cubic fourfolds.

Background

The introduction notes that only a few nontrivial birational maps between non-isomorphic irrational cubic fourfolds are known. Although the paper constructs such a birational pair for very general cubic fourfolds in the Hassett divisor C_44, the authors state that a general method for distinguishing birational classes of irrational cubic fourfolds remains unavailable.

References

However, it is still unclear how to distinguish the birational class of one irrational cubic fourfold from another.

— Birational cubic fourfolds via an Enriques Cremona transformation  (2609.10353 - Brooke et al., 9 Sep 2026) in Section 1, Introduction

Suppose $X$ and $Y$ are two cubic fourfolds with $Ku(X)\simeq Ku(Y)$. Then $X$ and $Y$ are birational.

— Birational cubic fourfolds via an Enriques Cremona transformation  (2609.10353 - Brooke et al., 9 Sep 2026) in Section 1, Introduction, Conjecture \ref{conj: Huy}

The authors conjecture that if $F(X)$ and $F(Y)$ are birational, then the underlying cubic fourfolds are also birational.

— Birational cubic fourfolds via an Enriques Cremona transformation  (2609.10353 - Brooke et al., 9 Sep 2026) in Section 1, Introduction, paragraph preceding Theorem \ref{thm: bir Fano}