Hilbert-Kunz density functions and F-thresholds

Abstract

The first author had shown earlier that for a standard graded ring R and a graded ideal I in characteristic p>0, with ℓ(R/I)<∞, there exists a compactly supported continuous function fR,I whose Riemann integral is the HK multiplicity eHK(R,I). We explore further some other invariants, namely the shape of the graph of fR,m (where m is the graded maximal ideal of R) and the maximum support (denoted as α(R,I)) of fR,I. In case R is a domain of dimension d ≥2, we prove that (R,m) is a regular ring if and only if fR,m has a symmetry fR,m(x)=fR,m(d−x), for all x. If R is strongly F-regular on the punctured spectrum then we prove that the F-threshold cI(m) coincides with α(R,I). As a consequence, if R is a two dimensional domain and I is generated by homogeneous elements of the same degree, then we have (1) a formula for the F-threshold cI(m) in terms of the minimum strong Harder-Narasimhan slope of the syzygy bundle and (2) a well defined notion of the F-threshold cI(m) in characteristic 0. This characterisation readily computes cI(n)(m), for the set of all irreducible plane trinomials k[x,y,z]/(h), where m=(x,y,z) and I(n)=(xn,yn,zn).