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 $VSP$.
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机译：我们显示出与Veronese表面非极性的三次四倍$ F $具有以下性质：其幂的总和$ VSP（F，10）$沿幂20的$ K3 $曲面是奇异的性曲线的总和。这涉及Mukai，Iliev和Ranestad的建筑。我们还证明了这些三次方在三次方的模量空间中形成一个除数，并且该除数不是Noether-Lefschetz除数。我们使用该结果证明非常普通的三次方与其$ VSP $之间没有平凡的Hodge对应关系。
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