Let
be a field of positive characteristic
and consider the ring
-
![{\displaystyle {}R=K[X,Y,Z]/{\left(X^{5}+Y^{3}+Z^{2}\right)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1777653199eab3e4cf096d233c61866085fb0a2f)
together with the ideal
and
.
Since
has a rational singularity, it is
-regular, i.e. all ideals are tightly closed. Therefore
and so the torsor
-
![{\displaystyle {}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)}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5da33be13d4227af041f4fbd5be274d6fa748c77)
is an affine scheme. In characteristic zero this can be proved by either using that
is a quotient singularity or by using the natural grading
(
)
where the corresponding cohomology class
gets degree
and then applying the geometric criteria on the corresponding projective curve
(rather the corresponding curve of the standard-homogenization
).