  We study questions surrounding cup-product maps which arise from pairs of
non-degenerate line bundles on an abelian variety. Important to our work is
Mumford's index theorem which we use to prove that non-degenerate line bundles
exhibit positivity analogous to that of ample line bundles. As an application
we determine the asymptotic behaviour of families of cup-product maps and prove
that vector bundles associated to these families are asymptotically globally
generated. To illustrate our results we provide several examples. For instance,
we construct families of cup-product problems which result in a zero map on a
one dimensional locus. We also prove that the hypothesis of our results can be
satisfied, in all possible instances, by a particular class of simple abelian
varieties. Finally, we discuss the extent to which Mumford's theta groups are
applicable in our more general setting.
