  The paper studies a rank 2 vector bundle on P1 x P3. Similarly to the
Horrocks - Mumford bundle on P4 this vector bundle encodes a lot of geometric
information. It is defined via the Serre construction by an abelian surface in
P1 x P3. The bundle is stable with respect to O(1,1), its jumping lines are
determined. Morover it is shown that it is closely related to a double structue
on P1 x P1 embedded in P1 x P3 as well as to a certain pencil of quadrics of
degree 4 in P3.
