On the closures of orbits of fourth order matrix pencils

We state a simple criterion for nilpotency of an $n\times n$ matrix pencil with respect to the action of $\operatorname{SL}_n(\mathbb C)\times \operatorname{SL}_n(\mathbb C) \times\operatorname{SL}_2(\mathbb C)$. We explicitly classify the orbits of matrix pencils for $n=4$ and describe the hierarchy of closures of nilpotent orbits. We also prove that the algebra of invariants of the action of $\operatorname{SL}_n(\mathbb C)\times \operatorname{SL}_n(\mathbb C)\times\operatorname{SL}_2(\mathbb C)$ on $\mathbb C_n\otimes\mathbb C_n\otimes\mathbb C_2$ is naturally isomorphic to the algebra of invariants of binary forms of degree $n$ with respect to the action of $\operatorname{SL}_2(\mathbb C)$.