اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOn polynomials with given Hilbert function and applications
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Using Macaulays correspondence we study the family of Artinian Gorenstein local algebras with fixed symmetric Hilbert function decomposition. As an application we give a new lower bound for cactus varieties of the third Veronese embedding. We discuss the case of cubic surfaces, where interesting phenomena occur.
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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 funct
ion}. 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 $kto infty$.
Given the Hilbert function $u$ of a closed subscheme of a projective space over an infinite field $K$, let $m_u$ and $M_u$ be, respectively, the minimum and the maximum among all the Castelnuovo-Mumford regularities of schemes with Hilbert function $
u$. I show that, for every integer $m$ such that $m_u leq m leq M_u$, there exists a scheme with Hilbert function $u$ and Castelnuovo-Mumford regularity $m$. As a consequence, the analogous algebraic result for an O-sequence $f$ and homogeneous polynomial ideals over $K$ with Hilbert function $f$ holds too. Although this result does not need any explicit computation, I also describe how to compute a scheme with the above requested properties. Precisely, I give a method to construct a strongly stable ideal defining such a scheme.
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 prove
s 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 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.
Hilbert-Kunz multiplicity and F-signature are numerical invariants of commutative rings in positive characteristic that measure severity of singularities: for a regular ring both invariants are equal to one and the converse holds under mild assumptio
ns. A natural question is for what singular rings these invariants are closest to one. For Hilbert--Kunz multiplicity this question was first considered by the last two authors and attracted significant attention. In this paper, we study this question, i.e., an upper bound, for F-signature and revisit lower bounds on Hilbert--Kunz multiplicity.
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