Hilbert-Kunz density function and asymptotic Hilbert-Kunz multiplicity for projective toric varieties

Abstract

For a toric pair $(X, D)$, where $X$ is a projective toric variety of dimension $d-1\geq 1$ and $D$ is a very ample $T$-Cartier divisor, we show that the Hilbert-Kunz density function $HKd(X, D)(\lambda)$ is the $d-1$ dimensional volume of ${\overline {\mathcal P}}_D \cap \{z= \lambda\}$, where ${\overline {\mathcal P}}_D\subset {\mathbb R}^d$ is a compact $d$-dimensional set (which is a finite union of convex polytopes). We also show that, for $k\geq 1$, the function $HKd(X, kD)$ can be replaced by another compactly supported continuous function $\varphi_{kD}$ which is `linear in $k$'. This gives the formula for the associated coordinate ring $(R, {\bf m})$: $$\lim_{k\to \infty}\frac{e_{HK}(R, {\bf m}^k) - e_0(R, {\bf m}^k)/d!}{k^{d-1}} = \frac{e_0(R, {\bf m})}{(d-1)!}\int_0^\infty\varphi_D(\lambda)d\lambda, $$ where $\varphi_D$ (see Proposition~1.2) is solely determined by the shape of the polytope $P_D$, associated to the toric pair $(X, D)$. Moreover $\varphi_D$ is a multiplicative function for Segre products. This yields explicit computation of $\varphi_D$ (and hence the limit), for smooth Fano toric surfaces with respect to anticanonical divisor. In general, due to this formulation in terms of the polytope $P_D$, one can explicitly compute the limit for two dimensional toric pairs and their Segre products. We further show that (Theorem~6.3) the renormailzed limit takes the minimum value if and only if the polytope $P_D$ tiles the space $M_{\mathbb R} = {\mathbb R}^{d-1}$ (with the lattice $M = {\mathbb Z}^{d-1}$). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.