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 F-F' of lines on the cubic F' is isomorphic to the Hilbert scheme Hilb(2) S of length two subschemes of S. The first main result of this paper is that Hilb(2) S parametrizes the variety V SP(F, 10) of presentations of the cubic form f, with F = Z(f), as a sum of 10 cubes, which yields an isomorphism between F-F' and V SP(F, 10). Furthermore, we show that V SP(F, 10) sets up a (6, 10) correspondence between F' and F-F'. The main result follows by a deformation argument. [References: 10]
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