Let $X$ be a curve (proper smooth variety of dimension $1$) over $\mathbf C$; $\mathcal L$ an invertible $\mathcal O_X$-module; $r = \dim_{\mathbf C}(\mathrm H^0(X; \mathcal L)) - 1$.

If $\mathcal L$ is very ample, then $\mathcal L(-P_1 - \cdots - P_{r - 1})$ is generated by global sections for $r - 1$ general points $P_1, \dots, P_{r - 1}$. In fact, let $i: X \to \mathbf P^r$ be the closed immersion corresponding to $\mathcal L$, then $\mathcal L(-P_1 - \cdots - P_{r - 1})$ is globally generated if and only if the $(r - 2)$-dimensional linear subspace of $\mathbf P^r$ passing through $P_1, \dots, P_{r - 1}$ does not pass through an $r$-th point of $X$, and this holds for general $r - 1$ points by the trisecant lemma.

I wonder whether $\mathcal L(-P_1 - \cdots - P_{r - 1})$ is generated by global sections for $r - 1$ general points $P_1, \dots, P_{r - 1}$ if $\mathcal L$ is only assumed to be globally generated.