line bundles $(L,M)$ satisfying condition and having nonzero index, there is no reason to expect that such maps should be nonzero. In fact, in §\[main:example\] we construct $2$-dimensional families of such cup-product problems which result in a zero map on a $1$-dimensional locus. In §\[theta:example\], we discuss an approach, used by Mumford, to study cup-product maps arising from pairs of ample line bundles. We discuss the extent to which these techniques are applicable in our more general situation.

Acknowledgments
---------------

I thank my Ph.D. advisor Mike Roth for suggesting the problem to me, for many profitable conversations, and for providing feedback which enabled me to improve the exposition of this work. I also benefited from helpful conversations with Greg Smith, Ernst Kani, Valdemar Tsanov, and A.T. Huckleberry. This work was conducted while I was a Ph.D. student at Queen’s University, where I was financially supported by several Ontario Graduate Scholarships.

Other notation and conventions
------------------------------

Throughout (unless explicitly stated otherwise) all abelian varieties are defined over a fixed algebraically closed field $\mathbb{K}$ of arbitrary characteristic.

If $E$ is a vector bundle on a $g$-dimensional abelian variety $X$, then we let ${\operatorname{c}}_{\ell}(E)$ denote its $\ell$th Chern class in the group ${\operatorname{A}}^{\ell}(X) 