= Restriction of the theta divisor to an Abel curve
{title2=$\phi_D^*\Theta=D_0$}
For a degree-$g$ <divisor class> $D$ with $h^0(\mathcal O_C(D))=1$, let $D_0$ be its unique effective representative. Under $\phi_D(P)=[D-P]$, evaluation of the unique <global section> gives $\phi_D^*\Theta=D_0$, including multiplicities. If $h^0(\mathcal O_C(D))\ge2$, every evaluation kernel is nonzero and the entire curve lies in the <theta divisor>. These restrictions also show that $h^0(\mathcal O(\Theta))=1$: after pulling back to a dominant product of curves, the ratio of any section to the defining section of the <theta divisor> is constant on each general coordinate curve, hence constant everywhere.
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