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

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.

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 $$\pi (Q)=\pi (P)$$.