$F$-thresholds $c^I(\mathbf{m})$ for projective curves

We show that if $R$ is a two dimensional standard graded ring (with the graded maximal ideal $\mathbf{m}$) of characteristic $p>0$ and $I\subset R$ is a graded ideal with $\mathrm{\ell }(R\mathrm{/}I)<\mathrm{\infty }$ then the $F$-threshold $c^I(\mathbf{m})$ can be expressed in terms of a strong HN (Harder-Narasimahan) slope of the canonical syzygy bundle on $ProjR$. Thus $c^I(\mathbf{m})$ is a rational number. This gives us a well defined notion, of the $F$-threshold $c^I({\bf m})$incharacteristic$0$,intermsofaHNslopeofthesyzygybundleon$\mbox{Proj}~R$. This generalizes our earlier result (in [TrW]) where we have shown that if $I$ has homogeneous generators of the same degree, then the $F$-threshold $c^I(\mathbf{m})$ is expressed in terms of the minimal strong HN slope (in char $p$) and in terms of the minimal HN slope (in char $0$), respectively, of the canonical syzygy bundle on $ProjR$. Here we also prove that, for a given pair $(R,I)$ over a field of characteristic $0$, if $({\mathbf{m}}_p,I_p)$ is a reduction mod $p$ of $(\mathbf{m},I)$ then $c^{I_p}({\bf m}p) \neq c^I{\infty}({\bf m})$implies$c^{I_p}({\bf m}_p)$has$p$ in the denominator, for almost all $p$.