Let $E= \{y^{2}z = x^{3} - xz^{2}\}\subset \mathbb{P}^{2}_{k}$ be an elliptic curve over an algebraically closed field $k$. Let $P = [0:0:1]$ be a point and let $D$ be a Weil divisor correspons to the one point, i.e. $D = P$. How to find a corresponding Cartier divisor on $E$? Also, I want to show that two line bundles $\mathscr{O}(2D), \mathscr{O}(3D)$ are globally generated.
What I tried is to cover $E$ with three affine open sets $U_{0}, U_{1}, U_{2}$ and find appropriate rational functions $f_{0}, f_{1}, f_{2}$ so that the collection $\{(U_{i}, f_{i})\}$ gives the corresponding Cartier divisor for $D$. Here $U_{i}$s are standard cover of $\mathbb{P}^{2}$. However, if we put $f_{0} = f_{1} = 1$, then I can't find suitable $f_{2}$ which only has a simple zero at $P$, so I think I have to try other $f_{1}$ and $f_{2}$. But I stucked.