Picard number of the generic fiber of an abelian fibered hyperkaehler manifold

We shall show that the Picard number of the generic fiber, in the sense of scheme, of an abelian fibered hyperk"ahler manifold over the projective space is always one. We then give a few applications for the Mordell-Weil group. In particular, by deforming O’Grady’s 10-dimensional manifold, we construct an abelian fibered hyperk"ahler manifold of Mordell-Weil rank 20, which is the maximum possible among all known ones.