= Theta pullback zero-divisor identity
{title2=$\deg D_z=g,\qquad\operatorname{AJ}(D_z)=z+\kappa$}
If $p\mapsto\theta(u(p)-z)$ is not identically zero, its zero <divisor> $D_z$ has these properties. The logarithmic derivative changes by $-2\pi i\omega_j$ across the paired $b_j$ translation, giving zero count $g$. For a variation of $z$, the logarithm changes there by $2\pi i\delta z_j$; integrating by parts in the first-moment contour integral gives $\delta\operatorname{AJ}(D_z)=\delta z$. Thus the difference is constant. Weighted zero <divisors>, rather than a distinct-zero assumption, make the statement valid at collisions.
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