Theta pullback zero-divisor identity ($\mathrm{deg}{D}_z=g,\mathrm{AJ}(D_z)=z+\kappa$)

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 $\delta \mathrm{AJ}(D_z)=\delta z$. Thus the difference is constant. Weighted zero divisors, rather than a distinct-zero assumption, make the statement valid at collisions.