F-thresholds c(m) for projective curves

Abstract

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 cI(m) can be expressed in terms of a strong HN (Harder-Narasimhan) slope of the canonical syzygy bundle on ProjR. Thus cI(m) is a rational number. This gives us a well defined notion, of the F-threshold cI(m) in characteristic 0, in terms of a HN slope of the syzygy bundle on ProjR. This generalizes our earlier result (in [14]) where we have shown that if I has homogeneous generators of the same degree, then the F-threshold cI(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 (mp,Ip) is a reduction mod p of (m,I) then cIp(mp)≠c∞I(m) implies cIp(mp) has p in the denominator, for almost all p.