A remarkable moduli space of rank 6 vector bundles related to cubic surfaces

We study the moduli space ${\text{fM}}^s(6;3,6,4)$ of simple rank 6 vector bundles $\text{E}$ on ${\text{PP}}^3$ with Chern polynomial $1+3t+6t^2+4t^3$ 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 ${\text{PP}}^3$ having an even set of 56 nodes (cf. \cite{CaTo}). 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 ${\text{fA}}^0$ of the G.I.T. quotient space of the projective space $\text{fB}:=\{B\in \text{PP}(U^{\vee }\otimes W\otimes V^{\vee })\}$ of triple tensors of type $(3,3,4)$ by the natural action of $SL(W)\times SL(U)$. We give several constructions for these bundles, which relate them to cubic surfaces in 3-space ${\text{PP}}^3$ and to cubic surfaces in the dual space $({\text{PP}}^3{)}^{\vee }$. One of these constructions, suggested by Igor Dolgachev, generalizes to other types of tensors. Moreover, we relate the socalled {\em cross-product involution} for $(3,3,4)$-tensors, introduced in \cite{CaTo}, with the Schur quadric associated to a cubic surface in ${\text{PP}}^3$ and study further properties of this involution.