Very ample line bundle on a projective curve

Question

This is an example in Hartshorne. $X$ is the nonsingular cubic curve $y^2z = x^3 - xz^2$ in $\mathbb{P}_k^2$, and $\mathscr{L} = \mathscr{L}(P_0)$, where $P_0$ is the point $(0,1,0)$. He claims that $\mathscr{L}$ is ample because $\mathscr{L}(3P_0) \cong \mathscr{O}_X(1)$, but I don't see how this isomorphism holds.

Answer 1

Note that there is a section of $\mathcal O_X(1)$ such that the divisor of zeros of this section is $3P_0$ (see this post). Then use Exercise 14.2.E in Vakil's notes.