ample line bundles on $X$. Cup-product maps of the form $$\label{ample:cp:prod:intro}
{\mathrm{H}}^0(X,L) \otimes {\mathrm{H}}^0(X,M) \xrightarrow{\cup} {\mathrm{H}}^0(X, L \otimes M)$$ have been extensively studied because they are related to the syzygies of $X$ with respect to a suitable projective embedding. We now describe some parts of this story because it is related to what we do here.

The starting point is work of D. Mumford [@MumI], [@MumII], [@MumIII], and [@Mum:Quad:Eqns]. In these papers, Mumford proved a number of results related to the moduli of abelian varieties, the syzygies of abelian varieties, and theta functions. Two important features to Mumford’s approach were the theta group of a line bundle and the observation that cup-product maps of the form can be studied by considering families of cup-product maps parametrized by the dual abelian variety. We discuss some aspects to Mumford’s approach in more detail in §\[theta:example\].

Mumford’s work led to many additional results and successful generalizations. Notable examples include work of S. Koizumi [@Koi], G. Kempf [@KempfAbVar], G. Pareschi [@Par], and Pareschi-Popa [@Par:Popa], [@Par:Popa:II]. These generalizations were related to questions which can be phrased in terms of the property $N_p$ of M. Green and R. Lazarsfeld [@Green:Laz:Np], [@Laz p. 116].

For example, Lazarsfeld 