  We show that a cubic fourfold F that is apolar to a Veronese surface has the
property that its variety of power sums VSP(F,10) is singular along a K3
surface of genus 20. We prove that these cubics form a divisor in the moduli
space of cubic fourfolds and that this divisor is not a Noether-Lefschetz
divisor. We use this result to prove that there is no nontrivial Hodge
correspondence between a very general cubic and its VSP.
