All-dimensional Hessian irreducibility conjecture for smooth cubic hypersurfaces

Let $f\in\bK[x_0,\dots,x_n]$ define a smooth cubic hypersurface $X=V(f)\subset\bP^n$, and let $\cH_f$ be its Hessian hypersurface. Say that $f$ is of Thom–Sebastiani type if, after a change of coordinates, it can be written using two distinct sets of variables. All-dimensional Hessian criterion conjecture. The Hessian hypersurface $\cH_f$ is irreducible and normal if and only if $f$ is not of Thom–Sebastiani type, for every $n\geq2$. Theorem A establishes this equivalence for $n\leq5$. The conjecture asks whether the same characterization extends to every projective dimension, while the paper explicitly presents it as unresolved.