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

Abstract

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 $\wedge ^2(V_{n+2})$ 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 $n\ge 5$. 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 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. Based on Manivel and Mezzetti (Manuscr Math 117:319–331, 2005), a complete classification of 4-planes $H\in {\mathcal {V}}$ with the reducible characteristic surface $S_H$ is given.