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

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

${}R=K[X,Y,Z]/{\left(X^{5}+Y^{3}+Z^{2}\right)}\,$

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 ${}Z\notin (X,Y)^{*}$ and so the torsor

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

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 ($\operatorname {deg} _{}^{}{\left(X\right)}=6,\,\operatorname {deg} _{}^{}{\left(Y\right)}=10,\,\operatorname {deg} _{}^{}{\left(Z\right)}=15$) 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 ${}U^{30}+V^{30}+W^{30}=0$).