Tight closure/Graded ring/Inclusion bound via strong HN-filtration/Remark

A notion of strong semistabilty exists also on higher dimensional normal projective varieties (depending on a polarization). If ${}\operatorname {Syz} _{d}\,$ is the top-dimensional syzygy bundle for a homogeneous ideal ${}I\subseteq R$ and ${}m$ is such that the minimal slope of ${}\operatorname {Syz} _{d}(m)$ is nonnegative, then ${}R_{\geq m}\subseteq I^{*}$. It is however more difficult to determine the degree of this syzygy bundle and to obtain exclusion results. Even if this bundle is strongly semistable there is no known inclusion/exclusion degree bound.