\(K3\) surfaces of genus 8 and varieties of sums of powers of cubic fourfolds (Q2701665)

Let \(F\) be a general cubic fourfold in the projective space \({\mathbb P}^5\) over the complex field and let \(f\) be an equation for \(F\). One can write \(f\) as a linear combination of 10 cubic powers of linear forms (and not less), as an immediate dimension count suggests. The subsets \(Z=\{L_0,\dots,L_9\}\) of the dual space \(({\mathbb P}^5)^\vee\) such that \(F=a_0L_0^3+\dots+a_9L_9^3\) are precisely the elements of \(\text{Hilb}_{10}({\mathbb P}^5)^\vee\) which are ''apolar'' to \(F\), in the sense that if one reads homogeneous polynomials over \(({\mathbb P}^5)^\vee\) as differential operators on \(f\), then \(g(f)=0\) for all \(g\) in the ideal of \(Z\).NEWLINENEWLINENEWLINEThe authors prove that the variety \(VPS =\{Z\in \text{Hilb}_{10}({\mathbb P}^5)^\vee: Z\) is apolar to \(F\}\) is isomorphic to the Fano variety of lines contained in some other cubic \(F'\subset{\mathbb P}^5\).NEWLINENEWLINENEWLINEThis correspondence is particularly interesting when \(F\subset{\mathbb P}^5\) is the apolar cubic fourfold of a general K3 surface \(S\) of genus \(8\), naturally embedded in \({\mathbb P}^8\) by a generator of the Picard group. In this case it turns out that the cubic \(F'\) corresponding to \(F\) in the previous construction is the pfaffian cubic such that \(\text{Hilb}_2(S)\) is isomorphic to the set of lines in \(F'\).