Hilbert coefficients and depth of fiber cones

Castelnuovo-Mumford regularity of associated graded modules and fiber cones of filtered modules

We give bounds on the Castelnuovo-Mumford regularity of the associated graded module of an arbitrary good filtration and of its fiber cone. These bounds extend previous results of Rossi-Trung-Valla and Linh.

Cohen-Macaulay fiber cones

Cohen Macaulay property of fiber cones of ideals is characterized in terms of its Hilbert series. Hilbert series of fiber cones of ideals with minimal mixed multiplicity is calculated. It is proved that the fiber cone of an m-primary ideal I with minimal mixed multiplicity is Cohen-Macaulay if and only if its reduction number is atmost one. Hilbert series of fiber cones of ideals generated by quadratic sequences in standard graded rings is computed by deforming it to a face ring of a simplicial complex. Applications are given to defining ideals of monomial projective space curves lying on the …

Castelnuovo-Mumford Regularity of the Fiber Cone for good filtrations

In this paper we show that there is a close relationship between the invariants characterizing the homogeneous vanishing of the local cohomology of the Rees algebra and the associated graded ring for the good filtrations case. We obtain relationships between the Castelnuovo-Mumford regularity of the fiber cone, associated graded ring, Rees algebra and reduction number for the good filtrations case.

Cohen-Macaulay 纤维锥的定义理想

Defining ideals of Cohen-Macaulay fiber cones

本文针对交换 Noether 局部环中的理想，给出了 Cohen-Macaulay 纤维锥的定义理想的构造方法。

On the fiber cone of monomial ideals

We consider the fiber cone of monomial ideals. It is shown that for monomial ideals $I\subset K[x,y]$ of height $2$, generated by $3$ elements, the fiber cone $F(I)$ of $I$ is a hypersurface ring, and that $F(I)$ has positive depth for interesting classes of height $2$ monomial ideals $I\subset K[x,y]$, which are generated by $4$ elements. For these classes of ideals we also show that $F(I)$ is Cohen--Macaulay if and only if the defining ideal $J$ of $F(I)$ is generated by at most 3 elements. In all the cases a minimal set of generators of $J$ is determined.

On fiber cones of ${\mathfrak m}$-primary ideals

Two formulas for the multiplicity of the fiber cone $F(I)=\oplus_{n=0}^{\infty} I^n/\m I^n$ of an $\m$-primary ideal of a $d$-dimensional Cohen-Macaulay local ring $(R,\m)$ are derived in terms of the mixed multiplicity $e_{d-1}(\m | I),$ the multiplicity $e(I)$ and superficial elements. As a consequence, the Cohen-Macaulay property of $F(I)$ when $I$ has minimal mixed multiplicity or almost minimal mixed multiplicity is characterized in terms of reduction number of $I$ and lengths of certain ideals. We also characterize Cohen-Macaulay and Gorenstein property of fiber cones of $\m$-primary ide…

The Hilbert Coefficients of the fiber cone and the $a$-invariant of the associated graded ring

Let $(A,\m)$ be a Noetherian local ring with infinite residue field and let $I$ be an ideal in $A$ and let $F(I) = \oplus_{n \geq 0}I^n/\m I^n$ be the fiber-cone of $I$. We prove certain relations among the Hilbert coefficients of $F(I)$ when the $a$-invariant of the associated graded ring $G(I)$ is negative.

The Cone of Hilbert Functions in the non-standard graded Case

We describe the cone of Hilbert functions of artinian graded modules finitely generated in degree 0 over the polynomial ring R = k[x, y] with the non-standard grading deg(x) = 1 and deg(y) = n, where n is any natural number.

On the lengths of quotients of ideals and depths of fiber cones

Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an $\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study the depths of $F(I)$ under certain depth assumptions on $G(I)$ and length condition on quotients of powers of $I$ and $J$, namely $\sum_{n\geq0}λ(\mathfrak{m}I^{n+1}/\mathfrak{m}JI^n)$ and $\sum_{n\geq0}λ(\mathfrak{m}I^{n+1} \cap J/\mathfrak{m}JI^n)$.

Artin-Nagata properties, minimal multiplicities, and depth of fiber cones

Minimal values of multiplicities of ideals have a strong relation with the depth of blowup algebras. In this paper, we introduce the notion of Goto-minimal $j$-multiplicity for ideals of maximal analytic spread. In a Cohen-Macaulay ring, inspired by the work of S. Goto, A. Jayanthan, T. Puthenpurakal, and J. Verma, we study the interplay among this new notion, the notion of minimal $j$-multiplicity introduced by C. Polini and Y. Xie, and the Cohen-Macaulayness of the fiber cone of ideals satisfying certain residual assumptions. We are also able to provide a bound on the reduction number of ide…

The cones of Hilbert functions of squarefree modules

In this paper, we study different generalizations of the notion of squarefreeness for ideals to the more general case of modules. We describe the cones of Hilbert functions for squarefree modules in general and those generated in degree zero. We give their extremal rays and defining inequalities. For squarefree modules generated in degree zero, we compare the defining inequalities of that cone with the classical Kruskal-Katona bound, also asymptotically.

纤维积环与Nagata理想化的Hilbert-Kunz重数

Hilbert-Kunz Multiplicity of Fiber Product Rings and Nagata Idealizations

本文为纤维积环和Nagata理想化推导出Hilbert-Kunz重数的显式公式，并给出新界与结构结果。

Fiber cones of ideals with almost minimal multiplicity

Fiber cones of 0-dimensional ideals with almost minimal multiplicity in Cohen-Macaulay local rings are studied. Ratliff-Rush closure of filtration of ideals with respect to another ideal is introduced. This is used to find a bound on the reduction number with respect to an ideal. Rossi's bound on reduction number in terms of Hilbert coefficients is obtained as a consequence. Sufficient conditions are provided for the fiber cone of 0-dimensional ideals to have almost maximal depth. Hilbert series of such fiber cones are also computed.

Cohen-Macaulay fiber cones and defining ideal of Rees algebras of modules

Generic Bourbaki ideals were introduced by Simis, Ulrich and Vasconcelos to study the Cohen-Macaulay property of Rees algebras of modules. In this article we prove that the same technique can sometimes be used to investigate the Cohen-Macaulay property of fiber cones of modules and to study the defining ideal of Rees algebras. This is possible as long as the Rees algebra of a given module $E$ is a deformation of the Rees algebra of a generic Bourbaki ideal $I$ of $E$. Our main technical result provides a deformation condition that in fact extends the applicability of generic Bourbaki ideals to…

Hilbert series of fiber cones of ideals of almost minimal mixed multiplicity

Let (R, m) be a Cohen-Macaulay local ring and I be an m-primary ideal. We introduce ideals of almost minimal mixed multiplicty which are analogues of ideals studied by J. Sally. The main theorem describes the Hilbert series of fiber cones of these ideals.

On the Hilbert function of the tangent cone of a monomial curve

In this paper we study the Hilbert function of $\gr_{\mathfrak{m}}(R)$, when $R$ is a numerical semigroup ring or, equivalently, the coordinate ring of a monomial curve. In particular, we prove a sufficient condition for a numerical semigroup ring in order get a non-decreasing Hilbert function, without making any assumption on its embedding dimension; moreover, we show how this new condition allows to improve known results about this problem. To this aim we use certain invariants of the semigroup, with particular regard to its \Apery-set.

4-Generated Pseudo Symmetric Monomial Curves with not Cohen-Macaulay Tangent Cones

In this article, standard bases of some toric ideals associated to 4-generated pseudo symmetric semigroups with not Cohen-Macaulay tangent cones at the origin are computed. As the tangent cones are not Cohen-Macaulay, non-decreasingness of the Hilbert function of the local ring was not guaranteed. Therefore, using these standard bases, Hilbert functions are explicitly computed as a step towards the characterization of Hilbert function. In addition, when the smallest integer satisfying $k(α_2+1)<(k-1)α_1+(k+1)α_{21}+α_3$ is $1$, it is proved that the Hilbert function of the local ring is non-de…

Fiber cones of rational normal scrolls are Cohen-Macaulay

In this short paper, we show that the fiber cones of rational normal scrolls are Cohen-Macaulay. As an application, we compute their Castelnuovo-Mumford regularities and $\mathbf{a}$-invariants, as well as the reduction number of the defining ideals of the rational normal scrolls. We also characterize the Gorensteinness of the fiber cone.

On the Gorenstein Property of the Fiber Cone to Filtrations

Let $(A, \mathfrak{m})$ be a Noetherian local ring and $\mathfrak{F}=(I_{n})_{n\geq 0}$ a filtration. In this paper, we study the Gorenstein properties of the fiber cone $F(\mathfrak{F})$, where $\mathfrak{F}$ is a Hilbert filtration. Suppose that $F(\mathfrak{F})$ and $G(\mathfrak{F})$ are Cohen-Macaulay. If in addition, the associated graded ring $G(\mathfrak{F})$ is Gorenstein; similarly to the $I$-adic case, we obtain a necessary and sufficient condition, in terms of lengths and minimal number of generators of ideals, for Gorensteiness of the fiber cone. Moreover, we find a description of …

Fiber cones, analytic spreads of the canonical and anticanonical ideals and limit Frobenius complexity of Hibi rings

Let ${\cal R}_{\mathbb{K}}[H]$ be the Hibi ring over a field $\mathbb{K}$ on a finite distributive lattice $H$, $P$ the set of join-irreducible elements of $H$ and $ω$ the canonical ideal of ${\cal R}_{\mathbb{K}}[H]$. We show the powers $ω^{(n)}$ of $ω$ in the group of divisors $\mathrm{Div}({\cal R}_{\mathbb{K}}[H])$ is identical with the ordinal powers of $ω$, describe the $\mathbb{K}$-vector space basis of $ω^{(n)}$ for $n\in\mathbb{Z}$. Further, we show that the fiber cones $\bigoplus_{n\geq 0}ω^n/\mathfrak{m}ω^n$ and $\bigoplus_{n\geq0}(ω^{(-1)})^n/\mathfrak{m}(ω^{(-1)})^n$ of $ω$ and $ω…

Cones of Hilbert functions

We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the cones generated by the Hilbert functions of all modules, all modules with bounded a-invariant, and all modules with bounded Castelnuovo-Mumford regularity. The first of these cones is infinite-dimensional and simplicial, the second is finite-dimensional but neither simplicial nor polyhedral, and the third is finite-dimensional and simplicial.

Toric ideals which are determinantal

Given any equigenerated monomial ideal $I$ with the property that the defining ideal $J$ of the fiber cone $ F(I)$ of $I$ is generated by quadratic binomials, we introduce a matrix such that the set of its binomial $2$-minors is a generating set of $J$. In this way, we characterize the fiber cone of sortable and Freiman ideals.

关于某些4生成伪2对称单项式曲线的切锥的Hilbert级数

On the Hilbert Series of the tangent cones for some 4-generated pseudo 2 symmetric monomial curves

本文研究了某些4生成伪对称单项式曲线的非Cohen-Macaulay切锥的Hilbert级数，通过显式计算证明了Hilbert函数的非递减性，并计算了相关环面理想的标准基。

Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

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…

Challenging computations of Hilbert bases of cones associated with algebraic statistics

In this paper we present two independent computational proofs that the monoid derived from $5\times 5\times 3$ contingency tables is normal, completing the classification by Hibi and Ohsugi. We show that Vlach's vector disproving normality for the monoid derived from $6\times 4\times 3$ contingency tables is the unique minimal such vector up to symmetry. Finally, we compute the full Hilbert basis of the cone associated with the non-normal monoid of the semi-graphoid for $|N|=5$. The computations are based on extensions of the packages LattE-4ti2 and Normaliz.

Hilbert's fourteenth problem over finite fields, and a conjecture on the cone of curves

We give examples over arbitrary fields of rings of invariants that are not finitely generated. The group involved can be as small as three copies of the additive group, as in Mukai's examples over the complex numbers. The failure of finite generation comes from certain elliptic fibrations or abelian surface fibrations having positive Mordell-Weil rank. Our work suggests a generalization of the Morrison-Kawamata cone conjecture from Calabi-Yau varieties to klt Calabi-Yau pairs. We prove the conjecture in dimension 2 in the case of minimal rational elliptic surfaces.

Blow-up algebras of secant varieties of rational normal scrolls

In this paper, we are mainly concerned with the blow-up algebras of the secant varieties of balanced rational normal scrolls. In the first part, we give implicit defining equations of their associated Rees algebras and fiber cones. Consequently, we can tell that the fiber cones are Cohen--Macaulay normal domains. Meanwhile, these fiber cones have rational singularities in characteristic zero, and are $F$-rational in positive characteristic. The Gorensteinness of the fiber cones can also be characterized. In the second part, we compute the Castelnuovo--Mumford regularities and $\mathbf{a}$-inva…

The cone spanned by maximal Cohen-Macaulay modules and an application

The aim of this paper is to define the notion of the Cohen-Macaulay cone of a Noetherian local domain R and to present its application to the theory of Hilbert-Kunz functions. It has been shown in Kurano's paper "Numerical equivalence defined on Chow groups of Noetherian local rings", Invent. Math. (2004), that, with a mild condition on R, the numerical Grothendieck group is a finitely generated torsion-free abelian group. The Cohen-Macaulay cone of R is a cone in the numerical Grothendieck group spanned by cycles represented by maximal Cohen-Macaulay modules. We study basic properties on the …

关于4生成伪对称单项式曲线的Hilbert函数单调性

On the monotonicity of the Hilbert functions for 4-generated pseudo-symmetric monomial curves

本文证明4生成伪对称数值半群局部环的Hilbert函数在生成元递增时非递减，通过刻画非Cohen-Macaulay切锥的标准基并显式计算Hilbert函数解决猜想。

拼接数值半群的切锥

Tangent Cones of Concatenated Numerical Semigroups

本文研究等差数列拼接数值半群的原点切锥与 Hilbert 级数，证明除对称类外各拼接类均为 Cohen–Macaulay，而对称类满足 Rossi 猜想。

Some geometric properties of Hilbert schemes of space curves

Let $H$ be the Hilbert scheme of curves in complex projective $3$-space, with $d\geq 3$ and genus $g \leq (d-2)^2/4$. A complete, explicit description of the cone of curves and the ample cone of $H$ is given. From this, partial results on the group $\mathop{Aut}(H)$ are deduced.

fiber-full scheme的局部研究

A local study of the fiber-full scheme

本文研究fiber-full scheme（推广Hilbert scheme的精细模空间）的局部性质，给出上同调在Gröbner退化下不变的充分条件，并建立其类比于Hilbert scheme的切障碍理论。

二维正规局部环上的Rees代数与除子滤过的约化纤维锥

Rees Algebras and the reduced fiber cone of divisorial filtrations on two dimensional normal local rings

本研究针对二维正规优局部环上的除子滤过，显式刻画其Rees代数的Proj概型与约化例外纤维结构，给出F. Russo定理的新证明，并明确解析展延不同取值对应的纤维性质。

素理想图的边理想：普通幂、纤维锥和线性幂

Edge Ideals of Prime Ideal Graphs over Finite Rings: Ordinary Powers, Fiber Cones, and Linear Powers

研究素理想图的结构及其边理想性质，确定最小顶点覆盖、不可约素分解，并分析普通幂的生成元和Hilbert函数，证明其为正常Cohen-Macaulay仿射半群环。

The power of pyramid decomposition in Normaliz

We describe the use of pyramid decomposition in Normaliz, a software tool for the computation of Hilbert bases and enumerative data of rational cones and affine monoids. Pyramid decomposition in connection with efficient parallelization and streamlined evaluation of simplicial cones has enabled Normaliz to process triangulations of size $\approx 5\cdot 10^{11}$ that arise in the computation of Hilbert series related to combinatorial voting theory.