Line bundle $\mathcal L(-P_1 - \cdots - P_{r - 1})$ on a curve being globally generated for $r - 1$ general points

Question

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.

Answer 1

No, this is not true. Take for instance a smooth plane curve $C$, and a double covering $\pi :X\rightarrow C$ branched along $k$ points, with $k> \frac{1}{2}\deg(C) $. Put $\mathscr{L}:=\pi ^*\mathscr{O}_C(1)$. Then $H^0(X,\mathscr{L})=\pi ^*H^0(C,\mathscr{O}_C(1))$, so the map $X\rightarrow \mathbb{P}^2$ defined by $\mathscr{L}$ factors through $\pi $. For any point $P$ of $X$, the linear system $\lvert \mathscr{L}(-P)\rvert$ has a fixed point, namely the point $Q\neq P$ such that $\pi (Q)=\pi (P)$.