E 8-Gleichung/x nicht in (y,z)^*/Erzwingende Algebra/Beispiel/en

Let $K$ be a field of positive characteristic  $p\ge 7$  and consider the ring

$R=K[X,Y,Z]/(X^5+Y^3+Z^2)$

together with the ideal  $I=(X,Y)$  and  $f=Z$.  Since $R$ has a rational singularity, it is $F$-regular, i.e. all ideals are tightly closed. Therefore  $Z\notin (X,Y{)}^{\ast }$  and so the torsor

$D(X,Y)\subseteq \mathrm{Spec}(K[X,Y,Z,S,T]/(X^5+Y^3+Z^2,SX+TY-Z))$

is an affine scheme. In characteristic zero this can be proved by either using that $R$ is a quotient singularity or by using the natural grading (${\mathrm{deg}}_{}^{}\left(X\right)=6,{\mathrm{deg}}_{}^{}\left(Y\right)=10,{\mathrm{deg}}_{}^{}\left(Z\right)=15$) where the corresponding cohomology class $\frac{Z}{XY}$ gets degree $-1$ and then applying the geometric criteria on the corresponding projective curve (rather the corresponding curve of the standard-homogenization $U^{30}+V^{30}+W^{30}=0$).