A Remarkable Moduli Space of Rank 6 Vector Bundles Related to Cubic Surfaces

We study the moduli space M(6; 3, 6, 4) of simple rank 6 vector bundles E on P with Chern polynomial 1 + 3t + 6t+4t and properties of these bundles, especially we prove some partial results concerning their stability. We first recall how these bundles are related to the construction of sextic nodal surfaces in P having an even set of 56 nodes (cf. [Ca-To]). We prove that there is an open set, corresponding to the simple bundles with minimal cohomology, which is irreducible of dimension 19 and bimeromorphic to an open set A of the G.I.T. quotient space of the projective space B := {B ∈ P(U ⊗ W ⊗ V )} of triple tensors of type (3, 3, 4) by the natural action of SL(W )× SL(U). We give several constructions for these bundles, which relate them to cubic surfaces in 3-space P and to cubic surfaces in the dual space (P). One of these constructions, suggested by Igor Dolgachev, generalizes to other types of tensors. Moreover, we relate the socalled cross-product involution for (3, 3, 4)-tensors, introduced in [Ca-To], with the Schur quadric associated to a cubic surface in P and study further properties of this involution.