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 c I ( m ) can be expressed in terms of a strong HN (Harder-Narasimhan) slope of the canonical syzygy bundle on Proj R . 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 Proj R . 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 Proj R . 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 .