  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 $n\leq 4$.
