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

It was shown in Ferapontov et al. (Lett Math Phys 108(6):1525–1550, 2018) 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 ∧ ²(V_n+2) such that the induced map Sym²H⟶ ∧ 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≥ 5. In this paper we address the most interesting borderline case n= 4. We prove that the variety V parametrizing those 4-planes H is an irreducible 38-dimensional PGL(V₆) -invariant subvariety of the Grassmannian G(4 , ∧ ²V₆). With every H∈ V we associate a characteristic cubic surface SH⊂ PH, the locus of rank 4 two-forms in H. We demonstrate that the induced characteristic map σ: V/ PGL(V₆) ⤏ M_c, where M_c denotes the moduli space of cubic surfaces in P³, is dominant, hence generically finite. Based on Manivel and Mezzetti (Manuscr Math 117:319–331, 2005), a complete classification of 4-planes H∈ V with the reducible characteristic surface S_H is given.