= Evaluation criterion for a prescribed principal divisor
{title2=$L(Q-P)=\ker\bigl(L(Q)\to\mathbb C^k\bigr)$}
Let $P=p_1+\cdots+p_k$ and $Q=q_1+\cdots+q_k$ be disjoint sums of distinct points on a <smooth projective curve>. A nonzero function with <principal divisor> $P-Q$ exists exactly when $L(Q-P)\ne0$: a nonzero member makes $(f)+Q-P$ an <effective divisor> of degree zero, forcing equality. Evaluation $f\mapsto(f(p_1),\ldots,f(p_k))$ on $L(Q)$ has <kernel of a linear map> $L(Q-P)$.
If $\ell(Q)=k$, a choice of <basis> $f_1,\ldots,f_k$ turns the criterion into $\det(f_j(p_i))=0$. This determinant is not identically zero. Starting with $L(Q)$, repeatedly choose a point where some nonzero function in the current space does not vanish; evaluation then lowers the <dimension> of that space by one. After $k$ choices the joint kernel is zero. Thus a generic tuple of points gives an invertible evaluation matrix, and no function with divisor $P-Q$. On a <genus one curve>, <Riemann-Roch theorem> gives $\ell(Q)=k$, so this obstruction applies for every $k>0$.
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