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

Abstract

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 4folds: 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 .TF' of lines on the cubic F' is isomorphic to the Hilbert scheme Hilb2 S of length two subschemes of S. The first main result of this paper is that Hilb2 S 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 NFF and VSP(F, 10). Furthermore, we show that VSP(F, 10) sets up a (6, 10) correspondence between F' and FF/ . The main result follows by a deformation argument. 1. PFAFFIAN AND APOLAR CUBIC 4-FOLDS ASSOCIATED TO K3 SURFACES OF GENUS 8 1.1. Let V be a 6-dimensional vector space over C. Fix a basis e,... ,e5 for V; then ei A ej for 0 i<j aijei Aej E A2V a skewsymmetric matrix M(g) = (aij), with aji = -aj. With Pliicker coordinates xij, the embedding of the Grassmannian G = G(2, V) in p14 = P(A2V) is then precisely the locus of rank 2 skewsymmetric 6 x 6 matrices / 0 X01 X02 X03 X04 X05 -x0i 0 X12 X13 X14 X15 M -X02 -X12 0 X23 X24 X25 -X03 -X13 -X23 0 X34 X35 -X04 -X14 -X24 -X34 0 X45 -X05 -X15 -X25 -X35 -X45 0 / Since the sum of two rank 2 matrices has rank at most 4, and any rank 4 skewsymmetric matrix is the sum of two rank 2 skewsymmetric matrices, the secant variety of G is the cubic hypersurface K defined by the 6 x 6 Pfaffian m of the matrix M. The dual variety of G in p14 = P(A2V*) is a cubic hypersurface K* = K (cf. [10]). K* is the secant variety of G* = G(V, 2), the Grassmannian of rank 2 quotient spaces of V, and of course G* C G. Received by the editors July 5, 1999. 2000 Mathematics Subject Classification. Primary 14J70; Secondary 14M15, 14N99. (2000 American Mathematical Society