I suppose the Picard group of a complete curve $C$ in $\mathbb P^n$ of degree $d>2$ is complicated. So, if we remove a general point $x\in C$ and denote $C'=C-x$, I think we have $\mathcal O_{C'}(1)\neq \mathcal O_{C'}$, here $\mathcal O_{C'}(1)$ is the restriction of $\mathcal O(1)$ on $\mathbb P^n$ to $C'$. I would like to know if the following is true:
Let $C_\lambda$ be a family of degree $d~(d>2)$ curves in $\mathbb P^n$ covering $\mathbb P^n$ and passing through some common $x$, and let $C_\lambda'=C_\lambda-x$. Is it true that there is at least one $\lambda$ such that $\mathcal O_{C_\lambda'}(1)\neq \mathcal O_{C_\lambda'}$?
Any comment or reference would be helpful. Thanks.