= Evaluation determinant on a product of curves
{title2=$\det\pi_*(L|_{\sum\Gamma_i})$}
On $C\times C^n$, let $\Gamma_i$ be the graph of the $i$th coordinate and $D=\sum_i\Gamma_i$. Successive restriction exact sequences give
$$
\det\pi_*(L|_D)\cong\bigotimes_i\operatorname{pr}_i^*L\otimes\mathcal O\left(-\sum_{i<j}\Delta_{ij}\right).
$$
When $h^0(L)=n$ and $h^1(L)=0$, the <determinant> of evaluation vanishes at tuples where $H^0(L(-D))$ is nonzero. On distinct points it is the ordinary evaluation <determinant>; the diagonal factors remove its forced zero when two points coincide. This explains why the formula remains valid at repeated points, using jets instead of repeated evaluations.
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