in §\[pair:index:condition:section\], allows the above described constructions of Pareschi-Popa to be generalized so as to apply more generally to this setting.

Main Problems
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The problems we consider are made possible by work of Mumford concerning the cohomology groups of non-degenerate line bundles on $X$. In more detail, let $L$ be a non-degenerate line bundle on $X$. Equivalently, $L$ is a line bundle on $X$ which has nonzero Euler characteristic. Mumford proved that there exists an integer ${\operatorname{i}}(L)$ with the property that $${\mathrm{H}}^j(X, L) = \begin{cases}
V_L & \text{for $j = {\operatorname{i}}(L)$; and} \\
0 & \text{for $j \not = {\operatorname{i}}(L)$,}
\end{cases}$$ where $V_L$ is the unique irreducible weight $1$ representation of the theta group of $L$, see for instance [@Mum §16 and p. 217], [@MumI Theorem 2, p. 297], and [@Sek:77 p. 726]. In addition, Mumford’s index theorem asserts that if $A$ is an ample line bundle on $X$, then the roots of the polynomial $P(N) := \chi(A^N \otimes L) $ are real and ${\operatorname{i}}(L)$ equals the number of positive roots counted with multiplicity, [@Mum p. 145].

To state our guiding problems let $L$ and $M$ be line bundles on $X$ which satisfy the condition that $$\label{intro:PIC}
\text{$\chi(L)$, $\chi(M)$, 