The Horrocks-Mumford bundle restricted to planes

We study the behavior of the Horrocks-Mumford bundle $F_{HM}$ when restricted to a plane $\mathbb{P}^2 \subset \mathbb{P}^4$, looking for all possible minimal free resolutions for the restricted bundle. To each of the 6 resolutions (4 stable and 2 unstable) we find, we then associate a subvariety of the Grassmannian $\mathbb{G}(2, 4)$ of planes in $\mathbb{P}^4$. We thus obtain a filtration of the Grassmannian, which we describe in the second part of this work.