Irrationality of a Very General Cubic Fourfold

This problem asks whether a very general smooth cubic hypersurface $X\subset\mathbb P^5_{\mathbb C}$ — one lying outside a countable union of proper Zariski-closed subsets of the parameter space of cubics — fails to be rational, i.e. is not birational to $\mathbb P^4$. The question became an explicit problem of birational classification around 1969, although no single first-proposal source is canonical.

What is known points in an intriguing direction. Certain special cubic fourfolds are rational, and Hassett identified divisors in the moduli space of cubics whose members carry Hodge structures related to K3 surfaces [Hassett2000SpecialCubic]. Kuznetsov proposed a derived-category counterpart, isolating a K3-type subcategory of $D^b(X)$ whose geometricity is conjecturally equivalent to rationality [Kuznetsov2010Cubic]; the interaction between the Hodge-theoretic and categorical pictures is surveyed by Huybrechts [Huybrechts2025K3Cubic].

Despite this rich structure theory, no known birational invariant distinguishes a very general cubic fourfold from a rational variety, and this is precisely the obstacle. Irrationality is widely expected, but a proof would require either a genuinely new invariant or, against expectation, a rationality construction valid generically. The problem remains open.