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

For a toric pair (X,D), where X is a projective toric variety of dimension d−1≥1 and D is a very ample T-Cartier divisor, we show that the Hilbert–Kunz density function HKd(X,D)(λ) is the d−1 dimensional volume of P‾_D∩{z=λ}, where P‾_D⊂R^d is a compact d-dimensional set (which is a finite union of convex polytopes). We also show that, for k≥1, the function HKd(X,kD) can be replaced by another compactly supported continuous function φ_kD which is ‘linear in k’. This gives the formula for the associated coordinate ring (R,m): limk→∞⁡[Formula presented]=[Formula presented]∫0∞φ_D(λ)dλ where φ_D (see Proposition 1.2) is solely determined by the shape of the polytope P_D, associated to the toric pair (X,D). Moreover φ_D is a multiplicative function for Segre products. This yields explicit computation of φ_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 renormalized limit takes the minimum value if and only if the polytope P_D tiles the space M_R=R^d−1 (with the lattice M=Z^d−1). As a consequence, one gets an algebraic formulation of the tiling property of any rational convex polytope.