Towards the classification of homogeneous third-order Hamiltonian operators

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(\Lambda^2V)$ that lie in the kernel of the natural map $S^2(\Lambda^2V)\to \Lambda^4V$. Non-equivalent operators correspond to different orbits of the natural action of $SL(n+1)$. Based on this result, we obtain a classification of such operators for $n\leq 4$.