I'm having trouble understanding a remark in Hartshorne: Let $X$ be the nonsingular projective cubic defined by $y^2z = x^3 - xz^2$ and put $P_0 = (0,1,0)$. The claim is that $\mathscr{L}(P_0)$ is not very ample because it is not generated by its global sections, and that this in turn is because $X$ is not rational, so $P_0$ cannot be equivalent to another point of $X$.
But why would $\mathscr{L}(P_0)$ being generated by its global sections imply $P_0$ is equivalent to another point? This is not clear to me. A less precise question: to what extent does this remark apply to $\mathscr{L}(P)$ for other points $P \in X$, or even other nonsingular projective curves of positive genus?