Let $k$ be a field, $X$ a smooth geometrically connected projective curve over $k$, and $\mathscr{L}$ an invertible sheaf on $X$. Then I think that $\mathscr{L}$ is algebraic equivalent to $\mathscr{O}_X$ if and only if $\deg \mathscr{L}= 0$. But I can't find references.
(two invertible sheaves $\mathscr{L}, \mathscr{N}$ are algebraic equivalent if for some $n \ge 1$ and for all $1 \le i \le n$, there exists a connected $k$-scheme of finite type $T_i$ and geometric points $t_i, s_i$ in $T_i$ with the same field, and an invertible sheaf $\mathscr{M}_i$ on $X_{T_i}$ such that $\mathscr{L}_{s_1} \cong \mathscr{M}_{1, s_1}, \mathscr{M}_{1, t_1} \cong \mathscr{M}_{2, s_2}, \dots , \mathscr{M}_{n, t_n} \cong \mathscr{N}_{t_n}$.)
The "only if" part is clear, since the degree is invariant under base changes. How can I show the "if" part? Or are there some references?
Using fppf descent, we may assume that $k$ is algebraically closed. And using Weil divisors, we may assume that $\mathscr{L} = \mathscr{O}_X(P - Q)$ for closed points $P, Q \in X$.
Thank you very much!