Hilbert-Kunz density functions and F-thresholds

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