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 $\mathbb P^1 \subset \mathbb P^4$?

  • $\begingroup$ Could you please explain what you mean by a vector bundle's 'splitting type'? $\endgroup$ – 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 $\mathcal{O}_{\mathbb{P}^1}(-1)\oplus\mathcal{O}_{\mathbb{P}^1}(6)$.

See for example this link and the references it gives.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.