On a class of spaces of skew-symmetric forms related to Hamiltonian systems of conservation laws

It was shown in \cite{FPV} that the classification of $n$-component systems of conservation laws possessing a third-order Hamiltonian structure reduces to the following algebraic problem: classify $n$-planes $H$ in $\wedge^2(V_{n+2})$ such that the induced map $Sym^2H\longrightarrow \wedge^4V_{n+2}$ has 1-dimensional kernel generated by a non-degenerate quadratic form on $H^*$. This problem is trivial for $n=2, 3$ and apparently wild for $n\geq 5$. In this paper we address the most interesting borderline case $n=4$. We prove that the variety $\mathcal{V}$ parametrizing those 4-planes $H$ is an irreducible 38-dimensional $PGL(V_6)$-invariant subvariety of the Grassmannian $G(4, \wedge^2V_6)$. With every $H\in\mathcal{V}$ we associate a {\it characteristic} cubic surface $S_H\subset \mathbb{P} H$, the locus of rank 4 two-forms in $ H$. We demonstrate that the induced characteristic map $\sigma: \mathcal{V} / PGL(V_6) \dashrightarrow \mathcal{M}_c,$ where $\mathcal{M}_c$ denotes the moduli space of cubic surfaces in $\mathbb{P}^3$, is dominant, hence generically finite. A complete classification of 4-planes $H\in\mathcal{V}$ with the reducible characteristic surface $S_H$ is given.