Density function for the second coefficient of the Hilbert–Kunz function on projective toric varieties

Abstract

We prove that, analogous to the HK density function, (used for studying the Hilbert-Kunz multiplicity, the leading coefficient of the HK function), there exists a β-density function g R,m : [0, ∞) −→ R, where (R, m) is the homogeneous coordinate ring associated to the toric pair (X, D), such thatwhere β(R, m) is the second coefficient of the Hilbert-Kunz function for (R, m), as constructed by Huneke-McDermott-Monsky Moreover we prove, (1) the function g R,m : [0, ∞) −→ R is compactly supported and is continuous except at finitely many points, (2) the function g R,m is multiplicative for the Segre products with the expression involving the first two coefficients of the Hilbert polynomials of the rings involved.Here we also prove and use a result (which is a refined version of a result by Henk-Linke) on the boundedness of the coefficients of rational Ehrhart quasi-polynomials of convex rational polytopes.

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

“…such that |c(λ n )| < C 2 , for some positive constant C 2 , independent of λ and n ∈ N [MT2,Lemma 33,Lemma 49]. Hence the lemma.…”

“…For a toric pair (X, D), a decomposition of C D = ∪ j F j was given in [MT1], (for d ≥ 3, as d = 2 corresponds to (P 1 , O P 1 (n)), for n ≥ 1, which is easy to handle directly), where F j 's are d-dimensional cones such that, each P j := F j ∩ P D is a convex rational polytope and is a closure of P ′ j := F j ∩ P D . In [MT2], the boundary of P D was studied and described in terms of the facets of P j 's. We recall the decomposition of C D and few properties of ∂(P D ) from [MT1] and [MT2] which are relevant for this work.…”

“…In [MT2], the boundary of P D was studied and described in terms of the facets of P j 's. We recall the decomposition of C D and few properties of ∂(P D ) from [MT1] and [MT2] which are relevant for this work.…”

“…The description of the HK-density function and the β-density function of the aasociated homogeneous coordinate ring (R, m) can be found in [T1], [MT1] and [MT2]. The facets of the polytope P D are given by the hyperplanes x = 0, y = 0, x = ay + c and y = d. We denote them by F 1 , F 2 , F 3 and F 4 respectively.…”

“…Similar to the HK density function, for a 'projectively normal' toric pair (X, D) (i.e., (X, D) is a toric pair such that the coordinate ring R is an integrally closed domain), it was shown in [MT2] that there exists a β-density function g R,m : [0, ∞) → R which similarly refine the β-invariant of [HMM]. More precisely, it was shown that the sequence of functions {g n (R, m) : [0, ∞) −→ R} n∈N , given by…”