The Horrocks-Mumford bundle and the Ferrand construction (Q1065084)

The Ferrand construction gives a multiplicity two structure on suitable smoothly embedded codimension two varieties. The authors show that in \({\mathbb{P}}^ 4\) the resulting non-reduced surfaces give rise to rank two vector bundles only if they are complete intersections, or a cubic scroll, or a quintic scroll. The only one giving a non-split vector bundle is the quintic scroll, which is obtained by embedding the symmetric square of an elliptic curve into \({\mathbb{P}}^ 4\). This gives another construction for the bundle of the title, and further insight into the intricate geometry of its sections.