F-thresholds c^I(m) for projective curves

We show that if R is a two dimensional standard graded domain (with the graded maximal ideal m) of characteristic p>0 and I⊂R is a graded ideal with ℓ(R/I)<∞, then the F-threshold c^I(m) can be expressed in terms of a strong HN (Harder-Narasimhan) slope of the canonical syzygy bundle on ProjR. Thus c^I(m) is a rational number. This gives us a well defined notion, of the F-threshold c^I(m) in characteristic 0, in terms of a HN slope of the syzygy bundle on ProjR. This generalizes our earlier result (in [13]) where we have shown that if I has homogeneous generators of the same degree, then the F-threshold c^I(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. In the present more general setting, the relevant slope may not be the minimal one. Here we also prove that, for a given pair (R,I) over a field of characteristic 0, if (m_p,I_p) is a reduction mod p of (m,I) then c^I_p(m_p)≠c_∞^I(m) implies c^I_p(m_p) has p in the denominator, for almost all p.