$K3$ surfaces of genus 8 and varieties of sums of powers of cubic fourfolds

The main result of this paper is that the variety of presentations of a general cubic form $f$ in $6$ variables as a sum of $10$cubes is isomorphic to the Fano variety of lines of a cubic $4$-fold $F'$, in general different from $F=Z(f)$.

A general $K3$ surface $S$ of genus $8$determines uniquely a pair of cubic $4$-folds: the apolar cubic $F(S)$ and the dual Pfaffian cubic $F'(S)$ (or for simplicity $F$ and $F'$). As Beauville and Donagi have shown, the Fano variety $\mathcal{F}_{F'}$ of lines on the cubic $F'$ is isomorphic to the Hilbert scheme $\operatorname{Hilb}_2S$ of length two subschemes of $S$. The first main result of this paper is that $\operatorname{Hilb}_2S$ parametrizes the variety $VSP(F,10)$ of presentations of the cubic form $f$, with $F=Z(f)$, as a sum of $10$ cubes, which yields an isomorphism between $\mathcal{F}_{F'}$ and $VSP(F,10)$. Furthermore, we show that $VSP(F,10)$ sets up a $(6,10)$ correspondence between $F'$ and $\mathcal{F}_{F'}$. The main result follows by a deformation argument.