nonzero and surjective for all $n{\geqslant}n_0$ and all $x\in X$. Equivalently, the vector bundle $$\label{vb-eqn}
{\mathrm{R}}^{{\operatorname{i}}(L\otimes M)}_{p_1*}(m^*(L^n \otimes E)\otimes p_2^*(M^n \otimes F))$$ is globally generated if $n {\geqslant}n_0$.

Our first approach to proving Theorem \[MainTheorem\] is logically independent of the main results of [@Par], [@Par:Popa], and [@Par:Popa:II].

In more detail, using Theorems \[vanishing:thm\] and \[Step2\], we prove that the cup-product maps are nonzero and surjective, for all $x \in X$, and all sufficiently large $n$. (See Proposition \[direct:cp:nonzero:surj\].) Applying Proposition \[mainprop1\], which generalizes [@Par Proposition 2.1], we deduce that the vector bundles are globally generated whenever $n$ is sufficiently large.

Our second approach to proving Theorem \[MainTheorem\] is to prove that the vector bundles are globally generated whenever $n$ is sufficiently large. We then deduce, using Proposition \[mainprop1\], that the cup-product maps are nonzero and surjective, for all points of $X$, and all sufficiently large $n$.

In our second approach, to prove that the vector bundles are globally generated, we apply Pareschi-Popa’s theory of $M$-regularity. Specifically, we apply [@Par:Popa Theorem 2.4, p. 289] to obtain a sufficient condition for such vector bundles to be globally generated. (See Proposition \[Par:Pop:glob:gen\].) In fact, we do not need the full strength of their 