Tight closure/Arithmetic deformation/Brenner Katzman/Example

Consider ${}\mathbb {Z} [x,y,z]/{\left(x^{7}+y^{7}+z^{7}\right)}$ and take the ideal ${}I={\left(x^{4},y^{4},z^{4}\right)}$ and the element ${}f=x^{3}y^{3}$. Consider reductions ${}\mathbb {Z} \rightarrow \mathbb {Z} /(p)$. Then

$f\in I^{*}{\text{ holds in }}\mathbb {Z} /(p)[x,y,z]/(x^{7}+y^{7}+z^{7}){\text{ for }}p\equiv 3\!\!\!\mod 7$

and

$f\not \in I^{*}{\text{ holds in }}\mathbb {Z} /(p)[x,y,z]/(x^{7}+y^{7}+z^{7}){\text{ for }}p\equiv 2\!\!\!\mod 7.$

In particular, the bundle ${}\operatorname {Syz} {\left(x^{4},y^{4},z^{4}\right)}$ is semistable in the generic fiber, but not strongly semistable for any reduction ${}p\equiv 2\!\!\!\mod 7$. The corresponding torsor is an affine scheme for infinitely many prime reductions and not an affine scheme for infinitely many prime reductions.