= Invertible cup-product direction for a line bundle on a curve
{title2=$\xi\smile-:H^0(L)\xrightarrow{\sim}H^1(L)$}
Suppose a <line bundle> $L$ of degree $g-1$ on a <smooth projective curve> has $h^0(L)=h^1(L)=r$. Choose $r$ distinct points where evaluation is invertible for both $H^0(L)$ and $H^0(K_X\otimes L^{-1})$. Such choices exist simultaneously: each <determinant> defines a nonempty open subset of $X^r$, since independent sections cannot vanish at every point. Let $\xi\in H^1(\mathcal O_X)=H^0(K_X)^*$ be a sum of nonzero weighted evaluation functionals at these points. The <Serre duality> <matrix> of $\xi\smile-$ factors as the product of the two invertible evaluation <matrices> and an invertible diagonal <matrix>. Its <determinant> is nonzero. This gives the nonzero leading <determinant> needed for the <Riemann singularity theorem>.
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