On the uniqueness of the Horrocks-Mumford-bundle
In this paper the authors show that there is only one stable \(rank\quad 2\) bundle \({\mathcal F}\) on \({\mathbb{P}}^ 4\) with Chern classes \(c_ 1=-1\), \(c_ 2=4\) and \(H^ 2({\mathbb{P}}_ 4,{\mathcal F}(-1))=0\). The vanishing of \(H^ 2\) can be omitted according to a result announced by Lipschitz and to appear in a paper by W. Decker. The result is proved by studying the variety of unstable planes and showing that it is a smooth surface fibred over \({\mathbb{P}}_ 1\) by elliptic curves. The singular fibres are pentagons and these give rise to the standard monad on \({\mathbb{P}}^ 4\).
uniqueness of the Horrocks-Mumford-bundle
scientific article; zbMATH DE number 3964012
scientific article