Hilbert-Kunz density functions and $F$-thresholds

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