## Cartier Divisors

### Cartier Divisors

Hello I am stucked on this problem: let $\mathrm{char}k=0$ and let $X=\mathcal{V}(zy^2-x^2(x+z))\subset\mathbb{P}^2$ in homogeneous coordinates $x,y,z$. Let $o=[0:0:1]$, $r=[-1:0:1]$ and $s=[0:1:0]$ points of $X$ and let $U_1=X-\{o\}$ and $U_2=X-\{r,s\}$. I want to show that the Cartier Divisor $D=\{(U_1,\frac{x}{y}),(U_2,1)\}$ is not principal.

Proof: by contradiction, if $D$ is principal then exists $(X,f)\in [D]$ with $f\in k(X)^*$, so the functions $f\frac{y}{x},\frac{x}{y}\frac{1}{f}\in k[U_1]$ and the functions $f,\frac{1}{f}\in k[U_2]$. Now, let $g=f\frac{y}{x}$; $g$ is rational so $g=\frac{F}{G}$ with $F,G$ homogeneous polynomials on $x,y,z$ such that neither $F$ nor $G$ vanish on $U_1$ so $\mathcal{V}(F)\cap X=\mathcal{V}(G)\cap X=\{o\}$, how can I contradict the hypothesis? Thank you very much.

Proof: by contradiction, if $D$ is principal then exists $(X,f)\in [D]$ with $f\in k(X)^*$, so the functions $f\frac{y}{x},\frac{x}{y}\frac{1}{f}\in k[U_1]$ and the functions $f,\frac{1}{f}\in k[U_2]$. Now, let $g=f\frac{y}{x}$; $g$ is rational so $g=\frac{F}{G}$ with $F,G$ homogeneous polynomials on $x,y,z$ such that neither $F$ nor $G$ vanish on $U_1$ so $\mathcal{V}(F)\cap X=\mathcal{V}(G)\cap X=\{o\}$, how can I contradict the hypothesis? Thank you very much.

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