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Register a new userDensity function for the second coefficient of the Hilbert-Kunz function
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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 $beta$-density function $g_{R, {bf m}}:[0,infty)longrightarrow {mathbb R}$, where $(R, {bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$int_0^{infty}g_{R, {bf m}}(x)dx = beta(R, {bf m}),$$ where $beta(R, {bf m})$ is the second coefficient of the Hilbert-Kunz function for $(R, {bf m})$, as constructed by Huneke-McDermott-Monsky. Moreover we prove, (1) the function $g_{R, {bf m}}:[0, infty)longrightarrow {mathbb R}$ is compactly supported and is continuous except at finitely many points, (2) the function $g_{R, {bf m}}$ 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.
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For a pair $(M, I)$, where $M$ is finitely generated graded module over a standard graded ring $R$ of dimension $d$, and $I$ is a graded ideal with $ell(R/I) < infty$, we introduce a new invariant $HKd(M, I)$ called the {em Hilbert-Kunz density function}. In Theorem 1.1, we relate this to the Hilbert-Kunz multiplicity $e_{HK}(M,I)$ by an integral formula. We prove that the Hilbert-Kunz density function is additive. Moreover it satisfies a multiplicative formula for a Segre product of rings. This gives a formula for $e_{HK}$ of the Segre product of rings in terms of the HKd of the rings involved. As a corollary, $e_{HK}$ of the Segre product of any finite number of Projective curves is a rational number. As an another application we see that $e_{HK}(R, {bf m}^k) - e(R, {bf m}^k)/d!$ grows at least as a fixed positive multiple of $k^{d-1}$ as $kto infty$.
We prove the existence of HK density function for a pair $(R, I)$, where $R$ is a ${mathbb N}$-graded domain of finite type over a perfect field and $Isubset R$ is a graded ideal of finite colength. This generalizes our earlier result where one proves the existence of such a function for a pair $(R, I)$, where, in addition $R$ is standard graded. As one of the consequences we show that if $G$ is a finite group scheme acting linearly on a polynomial ring $R$ of dimension $d$ then the HK density function $f_{R^G, {bf m}_G}$, of the pair $(R^G, {bf m}_G)$, is a piecewise polynomial function of degree $d-1$. We also compute the HK density functions for $(R^G, {bf m}_G)$, where $Gsubset SL_2(k)$ is a finite group acting linearly on the ring $k[X, Y]$.
We had shown earlier that for a standard graded ring $R$ and a graded ideal $I$ in characteristic $p>0$, with $ell(R/I) <infty$, there exists a compactly supported continuous function $f_{R, I}$ whose Riemann integral is the HK multiplicity $e_{HK}(R, I)$. We explore further some other invariants, namely the shape of the graph of $f_{R, {bf m}}$ (where ${bf m}$ is the graded maximal ideal of $R$) and the maximum support (denoted as $alpha(R,I)$) of $f_{R, I}$. In case $R$ is a domain of dimension $dgeq 2$, we prove that $(R, {bf m})$ is a regular ring if and only if $f_{R, {bf m}}$ has a symmetry $f_{R, {bf m}}(x) = f_{R, {bf m}}(d-x)$, for all $x$. If $R$ is strongly $F$-regular on the punctured spectrum then we prove that the $F$-threshold $c^I({bf m})$ coincides with $alpha(R,I)$. As a consequence, if $R$ is a two dimensional domain and $I$ is generated by homogeneous elements of the same degree, thene have (1) a formula for the $F$-threshold $c^I({bf m})$ in terms of the minimum strong Harder-Narasimahan slope of the syzygy bundle and (2) a well defined notion of the $F$-threshold $c^I({bf m})$ in characteristic $0$. This characterisation readily computes $c^{I(n)}({bf m})$, for the set of all irreducible plane trinomials $k[x,y,z]/(h)$, where ${bf m} = (x,y,z)$ and $I(n) = (x^n, y^n, z^n)$.
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 $lim_{pto infty}e_{HK}(R_p, I_p)$ is equivalent, for any fixed $mgeq d-1$, to the existence of $lim_{pto infty}ell(R_p/I_p^{[p^m]})/p^{md}$. This we get as a consequence of Theorem 1.1: As $prightarrow 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 $mgeq d-1$. In particular, to define the HK density function $f^{infty}(R, I)$ in characteristic 0, it is enough to prove the existence of $lim_{pto infty} f_m(R_p, I_p)$, for any fixed $mgeq d-1$. 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.
For a toric pair $(X, D)$, where $X$ is a projective toric variety of dimension $d-1geq 1$ 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 ${overline {mathcal P}}_Dsubset {mathbb R}^d$ is a compact $d$-dimensional set (which is a finite union of convex polytopes). We also show that, for $kgeq 1$, 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 $(R, {bf m})$: $$lim_{kto infty}frac{e_{HK}(R, {bf m}^k) - e_0(R, {bf m}^k)/d!}{k^{d-1}} = frac{e_0(R, {bf m})}{(d-1)!}int_0^inftyvarphi_D(lambda)dlambda, $$ 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.
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