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

Abstract

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 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.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 0 , . . . , e 5 for V ; then e i ∧ e j for 0 ≤ i < j ≤ 5 form a basis for the Plücker space ∧ 2 V of 2-dimensional subspaces in V or lines in P 5 = P(V ). We associate to a 2-vector g = i<j a ij e i ∧ e j ∈ ∧ 2 V a skewsymmetric matrix M (g) = (a ij ), with a ji = −a ij . With Plücker coordinates x ij , the embedding of the Grassmannian G = G(2, V ) in P 14 = P(Λ 2 V ) is then precisely the locus of rank 2 skewsymmetric 6 × 6 matrices M =

On Waring's problem for several algebraic forms

“…It has a very rich geometry, see e.g. [6,16,18] for some old and new results. If n = 1, then it is an open subset of a projective space (see [2]), but much remains unknown for more than two variables.…”

Moduli of double EPW-sextics

“…By (7.2.26) we get that v 0 ∧ γ(0) = 0; a straightforward computation -see Table (11) -gives that γ(0) = 0 (recall that by hypothesis m 11 = 0). By (7.2.26) we get that v 0 ∧ γ(−1) = 0, this implies that γ(−1) = 0 -see Table (13). By (7.2.26) we get that v…”

“…Next recall that A c,L ∈ W ψ fix is sent to itself by the 1-PS λ F2 and hence A c,L decomposes as the direct sum of its weight subspaces: we let A c,L (i) ⊂ A c,L be the weight-i subspace (thus A c,L (i) is A c,L,2−i in the old notation). Tables (11), (12) and (13) give bases of A c,LM (i) for i = 0, ±1. A few explanations regarding notation: we denote v i ∧ v j ∧ v k by (ijk), we let ℓ j be the j-th element of the basis of L M given by (7.2.23).…”

“…a double cover of a particular kind of sextic hypersurface (an EPW-sextic, first introduced by Eisenbud-Popescu-Walter in [5]) and hence it has an explicit description. We should point out that the generic double EPW-sextic is not isomorphic (nor birational) to the Hilbert square of a K3 surface, and that only a handful of explicit locally complete families of hyperkähler varieties of dimension greater than 2 have been constructed, see [2,4,12,13,16] for the other families.…”

Contractions of hyper-Kähler fourfolds and the Brauer group

“…(2) 38 is a variety of sums of powers V SP (F, 10) where F is a cubic threefold in P 5 and points of V SP (F, 10) correspond to the ways of writing the equation of F as a sum of ten cubes of linear forms ([IR1] and [IR3]).…”