$\beta$-density function on the class group of projective toric varieties

We prove the existence of a compactly supported, continuous (except at finitely many points) function $g_{I, {\bf m}}: [0, \infty)\longrightarrow \mathbb{R}$ for all monomial prime ideals $I$ of $R$ of height one where $(R, {\bf m})$ is the homogeneous coordinate ring associated to a projectively normal toric pair $(X, D)$, such that $$\int_{0}^{\infty}g_{I, {\bf m}}(\lambda)d\lambda=\beta(I, {\bf m}),$$ where $\beta(I, {\bf m})$ is the second coefficient of the Hilbert-Kunz function of $I$ with respect to the maximal ideal ${\bf m}$, as proved by Huneke-McDermott-Monsky \cite{HMM2004}. Using the above result, for standard graded normal affine monoid rings we give a complete description of the class map $\tau_{{\bf m}}:\text{Cl}(R)\longrightarrow \mathbb{R}$ introduced in \cite{HMM2004} to prove the existence of the second coefficient of the Hilbert-Kunz function. Moreover, we show the function $g_{I, {\bf m}}$ is multiplicative on Segre products with the expression involving the first two coefficients of the Hilbert polynomial of the rings and the ideals. \keywords{coefficients of Hilbert-Kunz function\and projective toric variety\and Hilbert-Kunz density function\and $\beta$-density function\and monomial prime ideal of height one.}