VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A^6

Abstract: This paper is concerned with the geometry of the Gorenstein locus of the Hilbert scheme of $14$ points on $\mathbb{A}^6$. This scheme has two components: the smoothable one and an exceptional one. We prove that the latter is smooth and identify the intersection of components as a vector bundle over the Iliev-Ranestad divisor in the space of cubic fourfolds. The ninth secant variety of the triple Veronese reembedding lies inside the Iliev-Ranestad divisor. We point out that this secant variety is set-theoretically a codimension two complete intersection and discuss the degrees of its equations.