Projective geometry of homogeneous second order Hamiltonian operators

We prove the invariance of homogeneous second-order Hamiltonian operators under the action of projective reciprocal transformations. We establish a correspondence between such operators in dimension $n$ and $3$-forms in dimension $n + 1$. In this way we classify second order Hamiltonian operators using the known classification of $3$-forms in dimensions $\leq$ 9. Systems of first-order conservation laws that are Hamiltonian with respect to such operators are also explicitly found. The integrability of the systems is discussed in detail.