VSPs of cubic fourfolds and the Gorenstein locus of the Hilbert scheme of 14 points on A6

To a cubic fourfold one may associate a geometric object (a hyperkähler manifold) via the theory of VSP and an algebraic object (a finite Gorenstein algebra) via apolarity. We prove that the associated algebra is smoothable if and only if the fourfold lies on the Iliev-Ranestad divisor (which parameterizes certain cubics whose VSP is isomorphic to the Hilbert scheme of two points on a K3 surface). This bridge allows us to give a detailed description of the algebraic side, i.e., the Gorenstein locus of 14 points on A6 and also to identify the equation of the Iliev-Ranestad divisor as the unique degree 10 invariant of SL6. As our main technical tool, we develop a relative version of apolarity, building on ideas of Elias-Rossi.