This is an example from Hartshorne's Algebraic Geometry (II, 7.6.3):
Let $X$ be the nonsingular cubic curve $y^{2}z=x^{3}-xz^{2}$ in $\mathbb{P}^{2}$ (it is not a rational curve). Let $\mathscr{L}$ be the invertible sheaf of divisor $P_{0}$, where $P_{0}$ is the point $(0,1,0)$ and the line $z=0$ meets the curve at the divisor $3P_{0}$.
It says that "this $\mathscr{L}$ is not generated by global sections because it it were, then $P_{0}$ will be linearly equivalent to some other point $Q \in X$". Why is this true?
Suppose $\mathscr{L}$ is generated by $s_{i}$'s, if I choose a sheaf $\mathscr{K}$ generated by $s_{i}^{-1}$. Then $\mathscr{LK}$ is principal, but it is not necessary the associated sheaf of a single point.
Any help is appreciated!