Variety of Power Sums and Divisors in the Moduli Space of Cubic Fourfolds

Abstract

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 which is the variety of power sums of a sextic curve. This relates constructions of Mukai and Iliev and Ranestad. We also 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 $VSP$.