I have a variety $C_{0}\subset\mathrm{Spec}\mathbb{C}[M_{Y}]$ living in some $Y$ (which is toric) and I am guessing, that it is $\mathbb{P}^{1}$. So I have to find a birational function $f:C_0\rightarrow\mathbb{P}^{1}$ and a good choice would be the characters from the lattice $M_Y$, $\chi^{r}$.
Now I want to find out, under which circumstances $\chi^{r}$ is injective. So a good idea would be to write $div(\chi^{r})=D^{+}-D^{-}, D^{+},D^{-}\geq 0$ and to check the intersection product $C_0\cdot D^{+}=C_0\cdot D^{-}$. For injectivity, this should be 1 (correct?) and by Bezout both, the curve and $D^{+}$, should have degree 1.
But then that $C_0$ just sits linear in $Y$ and the case isn't interesting. Furthermore, we know that $V(xz-y^2)\cong\mathbb{P}^{1}$, but this curve has degree 2. So somewhere above should be a mistake or there is some ramification involved I don't see.
Thanks for your help
Richard