Tight closure/2 dim, standard-graded/interpretation with syzygzy bundles and extensions/Affineness characterization/description

We can also give a geometric interpretation of what it means for $f$ to belong to the tight closure of the ideal in terms of the torsor defined by $f$:

$f\in I^{*}$ if and only if $T={\mathbb {P} }({{\mathcal {S}}'}^{*})-{\mathbb {P} }({\mathcal {S}}^{*})$ is not an affine variety.

Moreover, $f\in I^{+}$ if and only if the torsor $T={\mathbb {P} }({{\mathcal {S}}'}^{*})-{\mathbb {P} }({\mathcal {S}}^{*})$ contains projective curves (geometric and cohomological properties).