# Horrocks-Mumford bundle

Let $E$ be the Horrocks-Mumford bundle, which is a rank-2 vector bundle on $\mathbb P^4$ with $c_1(E)=5$ and $c_2(E)=10$, defined by some combinatorical construction (see Okonek, Schneider, Schindler, Vector bundles on Complex Projective Spaces, 2.3), and having a section whose zero set is dimension-2 torus in $\mathbb P^4$. Could you help me to calculate its splitting type on all the lines $\mathbb P^1 \subset \mathbb P^4$?

• Could you please explain what you mean by a vector bundle's 'splitting type'? – Michael Albanese Dec 21 '15 at 6:48

The generic splitting type is $\mathcal{O}_{\mathbb{P}^1}(2)\oplus \mathcal{O}_{\mathbb{P}^1}(3)$.
However there are lines (jumping lines) on a rational fourfold where the type is $\mathcal{O}_{\mathbb{P}^1}(1)\oplus\mathcal{O}_{\mathbb{P}^1}(4)$, $\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}(5)$ or even $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(6)$.