Towards Hilbert-Kunz density functions in Characteristic $0$

Abstract: For a pair $(R, I)$, where $R$ is a standard graded domain of dimension $d$ over an algebraically closed field of characteristic $0$ and $I$ is a graded ideal of finite colength, we prove that the existence of $\lim_{p\to \infty}e_{HK}(R_p, I_p)$ is equivalent, for any fixed $m\geq d-1$, to the existence of $\lim_{p\to \infty}\ell(R_p/I_p^{[p^m]})/p^{md}$. This we get as a consequence of Theorem 1.1: As $p\rightarrow \infty $, the convergence of the HK density function $f{(R_p, I_p)}$ is equivalent to the convergence of the truncated HK density functions $f_m(R_p, I_p)$ (in $L^{\infty}$ norm) of the {\it mod $p$ reductions} $(R_p, I_p)$, for any fixed $m\geq d-1$. In particular, to define the HK density function $f^{\infty}(R, I)$ in characteristic 0, it is enough to prove the existence of $\lim_{p\to \infty} f_m(R_p, I_p)$, for any fixed $m\geq d-1$. This allows us to prove the existence of $e_{HK}^{\infty}(R, I)$ in many new cases, {\em e.g.}, when $\mbox{Proj~R}$ is a Segre product of curves, for example.