Globally generated vector bundles with $c_1=5$ on ${\mathbb{P}}^n$, $n\ge 4$

Abstract

We complete the classification of globally generated vector bundles with small $c_1$ on projective spaces by treating the case $c_1=5$ on ${\mathbb{P}}^n$, $n\ge 4$ (the case $c_1\le 3$ has been considered by Sierra and Ugaglia, while the cases $c_1=4$ on any projective space and $c_1=5$ on ${\mathbb{P}}^2$ and ${\mathbb{P}}^3$ have been studied in two of our previous papers). It turns out that there are very few indecomposable bundles of this kind: besides some obvious examples there are, roughly speaking, only the (first twist of the) rank 5 vector bundle which is the middle term of the monad defining the Horrocks bundle of rank 3 on ${\mathbb{P}}^5$, and its restriction to ${\mathbb{P}}^4$. We recall, in an appendix, from our preprint [arXiv:1805.11336], the main results allowing the classification of globally generated vector bundles with $c_1=5$ on ${\mathbb{P}}^3$. Since there are many such bundles, a large part of the main body of the paper is occupied with the proof of the fact that, except for the simplest ones, they do not extend to ${\mathbb{P}}^4$ as globally generated vector bundles.

v2: improvement of presentation. arXiv admin note: text overlap with arXiv:1805.11336

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