= Riemann vanishing theorem
{c}
{title2=$\{z:\theta(z,B)=0\}=W_{g-1}-\kappa$}
For the period matrix of a genus-$g$ <compact Riemann surface>, there is a vector $\kappa$ depending on the marking and base point such that the <Riemann theta function> vanishes exactly on this translate of the image of the <Abelian sum map>. Pairing sides of the cut surface proves that a nonzero pullback $\theta(u(p)-z)$ has a degree-$g$ zero <divisor> whose Abel sum is $z+\kappa$. At a theta zero that pullback includes the base point. Irreducibility and dimension of the effective-divisor locus give equality of supports; the primitive <First Chern class> of the theta <line bundle> makes the <divisor> reduced.
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