Tight closure/Not plus closure/Dimension two/Standard-graded/Fact/Proof

Proof

Consider

${}R={\mathbb {F} }_{2}(t)[x,y,z]/(g)\,.$

In this ring ${}y^{3}z^{3}\in I^{*}$, but it can not belong to the plus closure. Else there would be a curve morphism

$Y\longrightarrow C_{{\mathbb {F} }_{2}(t)}$

which annihilates the cohomology class ${}c$ and this would extend to a morphism of relative curves over ${}{\mathbb {A} }_{{\mathbb {F} }_{2}}^{1}$ almost everywhere.

To fact