A result of Watanabe and Yoshida says that an unmixed local ring of positive characteristic is regular if and only if its Hilbert-Kunz multiplicity is one. We show that, for fixed $p$ (characteristic) and $d$ (dimension), there exist a number $ε(d,p) > 0$ such that any nonregular unmixed ring $R$ has Hilbert-Kunz multiplicity at least $1+ε(d,p)$. We also show that local rings with sufficiently small Hilbert-Kunz multiplicity are Cohen-Macaulay and $F$-rational.
We show that the Hilbert-Kunz multiplicities of the reductions to positive characteristics of an irreducible projective curve in characteristic 0 have a well-defined limit as the characteristic tends to infinity.
We show that Bertini theorems hold for $F$-signature and Hilbert--Kunz multiplicity. In particular, if $X \subseteq \mathbb{P}^n$ is normal and quasi-projective with $F$-signature greater than $λ$ (respectively the Hilbert--Kunz multiplicity is less than $λ$) at all points $x \in X$, then for a general hyperplane $H \subseteq \mathbb{P}^n$ the $F$-signature (respectively Hilbert--Kunz multiplicity) of $X \cap H$ is greater than $λ$ (respectively less than $λ$) at all points $x \in X \cap H$.
We show that the Hilbert-Kunz multiplicity is a rational number for an R_+-primary homogeneous ideal I=(f_1, ..., f_n) in a two-dimensional graded domain R of finite type over an algebraically closed field of positive characteristic. Moreover we give a formula for the Hilbert-Kunz multiplicity in terms of certain rational numbers coming from the strong Harder-Narasimhan filtration of the syzygy bundle Syz(f_1, . . ., f_n) on the projective curve Y = Proj R.
Let $k$ be an algebraically closed field of characteristic $p > 0$. We show that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $λ$ at all points $x\in X$, then for a general hyperplane $H\subseteq\mathbb{P}^n_k$, the Hilbert--Kunz multiplicity of $X\cap H$ is less than $λ$ at all points $x\in X\cap H$. This answers a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker, whose conclusion is the same as ours when $X\subseteq\mathbb{P}^n_k$ is normal. In the process, we substantially generalize certain uniform estimate…
This paper develops a theory of equimultiplicity for Hilbert-Kunz multiplicity and uses it to study the behavior of Hilbert-Kunz multiplicity on the Brenner-Monsky hypersurface. A number of applications follows, in particular we show that Hilbert-Kunz multiplicity attains infinitely many values and that equimultiple strata may not be locally closed.
This paper concerns the question of whether a more direct limit can be used to obtain the limit Hilbert-Kunz multiplicity, a possible candidate for a characteristic zero Hilbert-Kunz multiplicity. The main goal is to establish an affirmative answer for one of the main cases for which the limit Hilbert-Kunz multiplicity is even known to exist, namely that of graded ideals in the homogeneous coordinate ring of smooth projective curves. The proof involves more careful estimates of bounds found independently by Brenner and Trivedi on the dimensions of the cohomologies of twists of the syzygy bundl…
We interpret Hilbert-Kunz theory of a graded ring of positive characteristic in terms of Frobenius asymptotic of cohomology of vector bundles on projective varieties. With this method we show that for almost all prime numbers there exist three-dimensional quartic hypersurface domains and modules of finite length with irrational Hilbert-Kunz multiplicity. From this we deduce that also the Hilbert-Kunz multiplicity of a local noetherian domain might be an irrational number.
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 invo…
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 assumptions. 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.
Hilbert-Kunz multiplicity of quadrics via Ehrhart theory
UCLA（加州大学洛杉矶分校）；Stockholm University（斯德哥尔摩大学）；BCAM – Basque Center for Applied Mathematics（巴斯克应用数学中心）；IKERBASQUE, Basque Foundation for Science（伊克尔巴斯基奎，巴斯克科学基金会）；College of Humanities and Sciences, Nihon University（日本大学人文科学部）
In this paper, we estimate the Hilbert-Kunz multiplicity of the (extended) Rees algebras in terms of some invariants of the base ring. Also, we give an explicit formula for the Hilbert-Kunz multiplicities of Rees algebras over Veronese subrings.
Let R denote a two-dimensional normal standard-graded domain over the algebraic closure K of a finite field of characteristic p, and let I denote a homogeneous primary ideal. We prove that the Hilbert-Kunz function of I has the form = e_{HK}(I) q^{2} + g (q) with rational Hilbert-Kunz multiplicity e_{HK}(I) and an eventually periodic function g (q).
We prove that the Hilbert-Kunz multiplicity is upper semi-continuous in F-finite rings and algebras of essentially finite type over an excellent local ring.
We define a function, called s-multiplicity, that interpolates between Hilbert-Samuel multiplicity and Hilbert-Kunz multiplicity by comparing powers of ideals to the Frobenius powers of ideals. The function is continuous in s, and its value is equal to Hilbert-Samuel multiplicity for small values of s and is equal to Hilbert-Kunz multiplicity for large values of s. We prove that it has an Associativity Formula generalizing the Associativity Formulas for Hilbert-Samuel and Hilbert-Kunz multiplicity. We also define a family of closures such that if two ideals have the same s-closure then they ha…
We describe the possible restrictions of the cotangent bundle Ω_{\PP^N} to an elliptic curve C \subset \PP^N. We apply this in positive characteristic to the computation of the Hilbert-Kunz function of a homogeneous R_+-primary ideal I \subset R in the graded section ring R = \bigoplus_{n \in \NN} Γ(C, Ø(n)).
We show that the Hilbert-Kunz multiplicity of a $d$-dimensional nonregular complete intersection over the algebraic closure of $F_p$, $p>2$ prime, is bounded by below by the Hilbert-Kunz multiplicity of the hypersurface $\sum _{i=0}^{d} x_i^2=0$, answering positively a conjecture of Watanabe and Yoshida in the case of complete intersections.
Let $R$ be a local ring of characteristic $p>0$ which is $F$-finite and has perfect residue field. We compute the generalized Hilbert-Kunz invariant for certain modules over several classes of rings: hypersurfaces of finite representation type, toric rings, $F$-regular rings.
In this paper we give new lower bounds on the Hilbert-Kunz multiplicity of unmixed non-regular local rings, bounding them uniformly away from one. Our results improve previous work of Aberbach and Enescu.
We establish the continuity of Hilbert-Kunz multiplicity and F-signature as functions from a Cohen-Macaulay local ring $(R,\m,k)$ of prime characteristic to the real numbers at reduced parameter elements with respect to the $\m$-adic topology.
We give bounds for the Hilbert-Kunz multiplicity of the product of two ideals, and we characterize the equality in terms of the tight closures of the ideals. Connections are drawn with $*$-spread and with ordinary length calculations.
We prove in a broad combinatorial setting, namely for finitely generated semipositive cancellative reduced binoids, that the Hilbert-Kunz multiplicity is a rational number independent of the characteristic.
We study Hilbert-Kunz multiplicity of non-singular curves in positive characteristic. We analyse the relationship between the Frobenius semistability of the kernel sheaf associated with the curve and its ample line bundle, and the HK multiplicity. This leads to a lower bound, achieved iff the kernel sheaf is Frobenius semistable, and otherwise to formulas for the HK multiplicity in terms of parameters measuring the failure of Frobenius semistability. As a byproduct, an explicit example of a vector bundle on a curve is given whose $n$-th iterated Frobenius pullback is not semistable, while its …
We prove that the Hilbert-Kunz function of the ideal $(I,It)$ of the Rees algebra $\mathcal{R}(I)$, where $I$ is an $\mathfrak{m}$-primary ideal of a $1$-dimensional local ring $(R,\mathfrak{m})$, is a quasi-polynomial in $e$, for large $e.$ For $s \in \mathbb{N}$, we calculate the Hilbert-Samuel function of the $R$-module $I^{[s]}$ and obtain an explicit description of the generalized Hilbert-Kunz function of the ideal $(I,It)\mathcal{R}(I)$ when $I$ is a parameter ideal in a Cohen-Macaulay local ring of dimension $d \geq 2$, proving that the generalized Hilbert-Kunz function is a piecewise p…
We determine the Hilbert-Kunz function of plane elliptic curves in odd characteristic, as well as over arbitrary fields the generalized Hilbert-Kunz functions of nodal cubic curves. Together with results of K. Pardue and P. Monsky, this completes the list of Hilbert-Kunz functions of plane cubics. Combining these results with the calculation of the (generalized) Hilbert-Kunz function of Cayley's cubic surface, it follows that in each degree and over any field of positive characteristic there are curves resp. surfaces taking on the minimally possible Hilbert-Kunz multiplicity.
We present results on the Watanabe-Yoshida conjecture for the Hilbert-Kunz multiplicity of a local ring of positive characteristic. By improving on a "volume estimate" giving a lower bound for Hilbert-Kunz multiplicity, we obtain the conjecture when the ring either has Hilbert-Samuel multiplicity less than or equal to five, or dimension less than or equal to six. For non-regular rings with fixed dimension, a new lower bound for the Hilbert-Kunz multiplicity is obtained.
In this paper, we ask the following question: what is the minimal value of the difference $e_{HK}(I) - e_{HK}(I')$ for ideals $I' \supseteq I$ with $l_A(I'/I) =1$? In order to answer to this question, we define the notion of minimal Hilbert-Kunz multiplicity for strongly F-regular rings. Moreover, we calculate this invariant for quotient singularities and for the coordinate ring of the Segre embedding: $P^{r-1} \times P^{s-1} \hookrightarrow P^{rs-1}$, respectively.
We prove that, under certain assumptions, generalized Hilbert-Kunz multiplicities can be expressed as linear combinations of classical Hilbert-Kunz multiplicities.
We develop Hilbert-Kunz theory in a combinatorial setting namely for binoids. We show that the Hilbert-Kunz multiplicity for commutative, finitely generated, semipositive, cancellative and reduced binoids exists and is a rational number. This implies that the corresponding Hilbert-Kunz multiplicity for the binoid algebras does not depend on the characteristic.
We compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals.
We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Trivedi. We describe the second coefficient as the limit of the Hilbert coefficients of Frobenius powers and show that it is additive in short exact sequences and satisfies a Northcott-type inequality.
Let k be an arbitrary field (of arbitrary characteristic) and let X = [x_{i,j}] be a generic m x n matrix of variables. Denote by I_2(X) the ideal in k[X] = k[x_{i,j}: i = 1, ..., m; j = 1, ..., n] generated by the 2 x 2 minors of X. We give a recursive formulation for the lengths of the k[X]-module k[X]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q)) as q varies over all positive integers using Grobner basis. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We give closed forms for the cases when m is at most 2, %as well as the closed forms …
The paper investigates the behavior of Hilbert-Samuel and Hilbert-Kunz multiplicities in families of ideals. It is shown that Hilbert-Samuel multiplicity is upper semicontinuous almost generally and that Hilbert-Kunz multiplicity is upper semicontinuous in families of finite type. Surprisingly, our machinery can be applied for families over Z and yields a partial solution to the question about characteristic zero Hilbert-Kunz multiplicity posed by Brenner, Li, and Miller. Another application is that for an affine ring the infimum in the definition of F-rational signature, an invariant defined …
Let $(R,\mathfrak m)$ be a local ring of characteristic $p>0$ and $M$ a finitely generated $R$-module. In this note we consider the limit: $\lim_{n\to \infty} \frac{\ell(H^0_{\mathfrak m}(F^n(M)))}{p^{n\dim R}} $ where $F(-)$ is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when $R$ is excellent and has isolated singularity. Furthermore, if $R$ is a complete intersection, then the limit is 0 if and only if the projective dimension of $M$ is less than the Krull dimension of $R$. We exploit this fact to give a quick proof that if $R$ is a comple…
In this note we first give a new bound on $e_{HK}(\sim)$ the Hilbert-Kunz multiplicity of invariant rings, by the help of the Noether's bound. Then, we simplify, extend and present applications of the reciprocity formulae due to L. Smith. He proved the formula over polynomial rings and his result is tight in the following sense: Over complete intersection rings with isolated singularity we show that the reciprocity formulae "$e_{HK}(R/I)+ e_{HK}(R/J) = e_{HK}(R/\underline{f})$" is equivalent with $\pd(I)<\infty$ when $I$ is an $\fm$-primary unmixed ideal linked to $J$ along with a regular sequ…
We discuss Hilbert-Kunz function from when it was originally defined to its recent developments. A brief history of Hilbert-Kunz theory is first recounted. Then we review several techniques involved in the study of Hilbert-Kunz functions by presenting some illustrative proofs without going into details of the technicalities. The second part of this article focuses on the Hilbert-Kunz function of an affine normal semigroup ring and relates it to Ehrhart quasipolynomials. We pay extra attention to its periodic behavior and discuss how the cellular decomposition constructed by Bruns and Gubeladze…
In the paper, by the singular Riemann-Roch theorem, it is proved that the class of the e-th Frobenius power can be described using the class of the canonical module for a normal local ring of positive characteristic. As a corollary, we prove that the coefficient of the second term of the Hilbert-Kunz function of a finitely generated A-module M vanishes if A is a Q-Gorenstein ring and M is of finite projective dimension. For a normal algebraic variety X over a perfect field of positive characteristic, it is proved that the first Chern class of the direct image of the structure sheaf via e-th Fr…
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 $HKd(X, D)(λ)$ is the $d-1$ dimensional volume of ${\overline {\mathcal P}}_D \cap \{z= λ\}$, 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 $φ_{kD}$ which is `linear in $k$'. This gives the formula f…
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 $g_{R, {\bf m}}:[0,\infty)\longrightarrow {\mathbb R}$, where $(R, {\bf m})$ is the homogeneous coordinate ring associated to the toric pair $(X, D)$, such that $$\int_0^{\infty}g_{R, {\bf m}}(x)dx = β(R, {\bf m}),$$ where $β(R, {\bf m})$ 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, \in…
Let p be a prime. The Hilbert-Kunz multiplicity, mu, of the element sum(x_i^(d_i)) of (Z/p)[x_1,..., x_s] depends on p in a complicated way. We calculate the limit of mu as p -> infinity. In particular when each d_i is 2 we show that the limit is 1 + the coefficient of z^(s-1) in the power series expansion of sec z + tan z.
In this paper, we investigate a lower bound (say $s_{HK}(p,d)$) on Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension $d$ with characteristic $p>0$. Especially, we focus three-dimensional local rings. In fact, as a main result, we will prove that $s_{HK}(p,3) = 4/3$ and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity 4/3 is isomorphic to the non-degnerate quadric hyperplanes $k[[X,Y,Z,W]]/(X^2+Y^2+Z^2+W^2)$ under mild conditions. Furthermore, we pose a generalization of the main theorem to the case of $\dim A \ge 4$ as a conjecture, and…
Let $R$ be an excellent Noetherian ring of prime characteristic. Consider an arbitrary nested pair of ideals (or more generally, a nested pair of submodules of a fixed finite module). We do \emph{not} assume that their quotient has finite length. In this paper, we develop various sufficient numerical criteria for when the tight closures of these ideals (or submodules) match. For some of the criteria we only prove sufficiency, while some are shown to be equivalent to the tight closures matching. We compare the various numerical measures (in some cases demonstrating that the different measures g…
In this thesis we compute the Hilbert-Kunz functions of two-dimensional rings of type ADE by using representations of their indecomposable, maximal Cohen-Macaulay modules in terms of matrix factorizations, and as first syzygy modules of homogeneous ideals. For these computations we need to compute the Hilbert-series of the involved first syzygy modules as well as the fact that the surface rings of type ADE defined over the complex numbers are the rings of invariants of C[x,y] under actions defined by the finite subgroups of SL2(C). Finally, we will show that the developed tools can be used to …
Suppose that h in F[x,y,z], char F=2, defines a nodal cubic. In earlier papers we made a precise conjecture as to the Hilbert-Kunz functions attached to the powers of h. Assuming this conjecture we showed that a class of characteristic 2 hypersurfaces has algebraic but not necessarily rational Hilbert-Kunz multiplicities. We now show that if the conjecture holds, then transcendental multiplicities exist, and in particular that the number \sum\binom{2n}{n}^{2}/(65,536)^{n}, proved transcendental by Schneider, is a Q-linear combination of Hilbert-Kunz multiplicities of characteristic 2 hypersurf…
The aim of this manuscript is to discuss the Hilbert-Kunz functions over an excellent local ring regular in codimension one. We study the shape of the Hilbert-Kunz functions of modules and discuss the properties of the coefficient of the second highest term in the function. Our results extend Huneke, McDermott and Monsky's result (Math. Res. Lett. 11 (2004), no. 4, 539-546) about the shape of the Hilbert-Kunz functions, and a theorem of the second author (J. Algebra 304 (2006), no. 1, 487-499) for rings with weaker conditions. In this paper, for a Cohen-Macaulay ring, we also explores an equiv…
Let I denote a homogeneous R_+-primary ideal in a two-dimensional normal standard-graded domain over an algebraically closed field of characteristic zero. We show that a homogeneous element f belongs to the solid closure I^* if and only if e_{HK}(I) = e_{HK}((I,f)), where e_{HK} denotes the (characteristic zero) Hilbert-Kunz multiplicity of an ideal. This provides a version in characteristic zero of the well-known Hilbert-Kunz criterion for tight closure in positive characteristic.
The Hilbert-Kunz multiplicity and $F$-signature are important invariants for researchers in commutative algebra and algebraic geometry. We provide software, and describe the automation of a calculation, for the two invariants in the case of intersection algebras over polynomial rings.
We present a unified approach to the study of Hilbert-Kunz multiplicity, F-signature, and related limits governed by Frobenius and Cartier linear actions in positive characteristic commutative algebra. We introduce general techniques that give vastly simplified proofs of existence, semicontinuity, and positivity. Furthermore, we give an affirmative answer to a question of Watanabe and Yoshida allowing the F-signature to be viewed as the infimum of relative differences in the Hilbert-Kunz multiplicites of the cofinite ideals in a local ring.
In this article, I give an iterative closed form formula for the Hilbert-Kunz function for any binomial hypersurface in general, over any feild of arbitrary positive characteristic. I prove that the Hilbert-Kunz multiplicity associated to any Binomial Hypersurface over any field of arbitrary positive characteristic is rational. As an example, I also prove the well known fact that for 1-dimensional Binomial Hypersurfaces, the Hilbert-Kunz multiplicity is a positive integer and give a precise account of the integer.
Suppose B=F[x,y,z]/h is the homogeneous coordinate ring of a characteristic p degree 3 irreducible plane curve C with a node. Let J be a homogeneous (x,y,z)-primary ideal and n -> e_n be the Hilbert-Kunz function of B with respect to J. Let q=p^n. When J=(x,y,z), Pardue (see R. Buchweitz, Q. Chen. Hilbert-Kunz functions of cubic curves and surfaces. J. Algebra 197 (1997). 246-267) showed that e_n=(7q^2)/3-q/3-R where R=5/3 if q is congruent to 2 (3), and is 1 otherwise. We generalize this, showing that e_n= (mu q^2) + (alpha q) - R where R only depends on q mod 3. We describe alpha and R in te…
