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

Source abstract

The main result of this paper is that the variety of presentations of a general cubic form f f in 6 6 variables as a sum of 10 10 cubes is isomorphic to the Fano variety of lines of a cubic 4 4 -fold F ′ F’ , in general different from F = Z ( f ) F=Z(f) . A general K 3 K3 surface S S of genus 8 8 determines uniquely a pair of cubic 4 4 -folds: the apolar cubic F ( S ) F(S) and the dual Pfaffian cubic F ′ ( S ) F’(S) (or for simplicity F F and F ′ F’ ). As Beauville and Donagi have shown, the Fano variety F F ′ \mathcal {F}_{F’} of lines on the cubic F ′ F’ is isomorphic to the Hilbert scheme Hilb 2 ⁡ S \operatorname {Hilb}_2S of length two subschemes of S S . The first main result of this paper is that Hilb 2 ⁡ S \operatorname {Hilb}_2S parametrizes the variety V S P ( F , 10 ) VSP(F,10) of presentations of the cubic form f f , with F = Z ( f ) F=Z(f) , as a sum of 10 10 cubes, which yields an isomorphism between F F ′ \mathcal {F}_{F’} and V S P ( F , 10 ) VSP(F,10) . Furthermore, we show that V S P ( F , 10 ) VSP(F,10) sets up a ( 6 , 10 ) (6,10) correspondence between F ′ F’ and F F ′ \mathcal {F}_{F’} . The main result follows by a deformation argument.