= Riemann singularity theorem
{c}
{title2=$\operatorname{mult}_L\Theta=h^0(X,L)$}
For a <compact Riemann surface> of genus $g\ge1$, the <theta divisor> in $\operatorname{Pic}^{g-1}(X)$ has the displayed multiplicity. A local <determinant description of the theta divisor> reduces to an $r$ by $r$ <matrix> vanishing at $L$, where $r=h^0(L)$. Its linear term is cup product with a tangent class in $H^1(\mathcal O_X)$. The <invertible cup-product direction for a line bundle on a curve> shows that its <determinant> has order exactly $r$. Consequently the <singular locus> consists precisely of classes with at least two independent <global sections>.
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