紧密封闭的理想的链条和家庭

弱分次理想族的重数

Multiplicities of weakly graded families of ideals

本文推广线性下有界弱分次理想族的重数概念，证明相关极限存在性、体积重数公式与Minkowski不等式，探究特定形式弱分次理想族的重数性质、不等式取等条件，推广Rees的结果并研究相关长度函数的渐近行为。

Mixed multiplicities of graded families of ideals

We show the existence (and define) the mixed multiplicities of arbitrary graded families of ideals under mild assumptions. In particular, our methods and results are valid for the case of arbitrary $\mathfrak{m}$-primary graded families. Furthermore, we provide a far-reaching "Volume = Multiplicity formula" for the mixed multiplicities of graded families of ideals.

分次理想族的度数函数

Degree functions of graded families of ideals

本文研究优正规局部环中分次理想族的重数与度数函数，将其表示为交积极限，并在二维情形下对除子滤过给出更精细结果及关于 Rees 赋值集合有限性与无限性的例子。

Families of cellular resolutions, their syzygies, and stability

We study families of cellular resolutions by looking at them as a category and applying tools from representation stability. We obtain sufficient conditions on the structure of the family to have a noetherian representation category and apply this to concrete examples of families. In the study of syzygies we make use of defining the syzygy module as a representation and find conditions for the finite generation of this representation. We then show that many families of cellular resolutions coming from powers of ideals satisfy these conditions and have finitely generated syzygies, including the…

关于理想的反向族的约化模与协约化模

Reduced and coreduced modules with respect to inverse families of ideals

本文针对交换环上理想的反向族，引入 Phi 约化模与协约化模，证明广义挠与完备化函子由最大真成员决定，并建立刻画、闭包性质及 Greenlees-May 型伴随，为理想族上的广义挠理论提供模论框架。

至多含两个参数理想族的单支曲线奇点

One branch curve singularities with at most 2-parameter families of ideals

给出了单支曲线奇点的理想至多形成2-参数族的判据。也就是说，我们提出了一个列表的平面曲线奇点从阿诺德的分类是最小的所有一个分支奇点有最多2-参数家庭的理想。

分次理想族的重数

Multiplicities Associated to Graded Families of Ideals

我们证明了与分次理想族相关的重数极限在非常一般的条件下存在。我们的大部分结果适用于解析非分歧等特征局部环，具有完美剩余域。我们给出了一些应用程序，包括一个“体积=多重性”公式，推广公式的Lazarsfeld和Mustata，并证明了Ulrich和Validashti的多重性存在作为一个限制理想在相当一般的环，包括分析的局部域。我们还证明了一个渐近的“加法公式”的限制的多重性，和公式限制增长的估值，这回答了一个问题所提出的作者，起亚Dalili和奥尔加Kashcheyeva。我们的证明的灵感来自奥肯科夫的哲学，计算限制的多重性的体积切片的适当的锥所产生的半群确定的适当的过滤上的一个家庭的代数对象。

牛顿-奥库诺夫体、雷斯代数与格雷德家族的单调理想解析度

Newton-Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

本文利用牛顿-奥库诺夫体研究格雷德单调理想家族的雷斯代数诺特性及解析度的组合解释，并探讨符号雷斯代数的生成类型与韦罗内塞次数。

On the Betti numbers of monomial ideals and their powers

Let $S=\mathbb{K}[x_1,\ldots,x_n]$ the polynomial ring over a field $\mathbb{K}$. In this paper for some families of monomial ideals $I \subset S$ we study the minimal number of generators of $I^k$. We use this results to find some other Betti numbers of these families of ideals for special choices of $n$, the number of variables.

$t$-clique ideal and $t$-independence ideal of a graph

In this paper, we introduce and study families of squarefree monomial ideals called clique ideals and independence ideals that can be associated to a finite graph. A family of clique ideals with linear resolutions has been characterized. Moreover some families of graphs for which the quotient ring of their clique ideal is Cohen-Macaulay are introduced and some homological invariants of the clique ideal of a graph $G$ which is the complement of a path graph or a cycle graph, are obtained. Also some algebraic properties of the independence ideal of path graphs, cycle graphs and chordal graphs ar…

Convex bodies and graded families of monomial ideals

We show that the mixed volumes of arbitrary convex bodies are equal to mixed multiplicities of graded families of monomial ideals, and to normalized limits of mixed multiplicities of monomial ideals. This result evinces the close relation between the theories of mixed volumes from convex geometry and mixed multiplicities from commutative algebra.

理想的分次族与凸区域

Graded families of ideals and convex regions

研究\(\mathbb{K}\)-域中理想分次族与相关凸区域（牛顿 - 奥昆科夫区域）的相互作用，结合阿图什 - 韦茨拓扑技术和凸几何性质，通过区域包含关系刻画一对理想分次族的渐近复苏数。

超越标准分次情形的理想的正规性：来自数值半群环的族

Normality of ideals beyond the standard graded setting: families from numerical semigroup rings

本文研究超越标准分次情形的理想正规性，从嵌入维数三的数值半群构造整闭准素理想，确定重数三或四时的生成元，并证明相应 Rees 代数为 Cohen–Macaulay 正规整环。

源于顶点可分解单纯复形的Rees代数

Normal Rees algebras arising from vertex decomposable simplicial complexes

本文证明了顶点可分解单纯复形的Rees代数是正规Cohen-Macaulay整环，并据此推导出无平方弱多拟阵理想及相关图覆盖理想和超图边理想的正规性。

Growth of multiplicities of graded families of ideals

Let $(R,\mathfrak{m})$ be a Noetherian local ring of dimension $d > 0$. Let $I_\bullet = \{I_n\}_{n \in \mathbb{N}}$ be a graded family of $\mathfrak{m}$-primary ideals in $R$. We examine how far off from a polynomial can the length function $\ell_R(R/I_n)$ be asymptotically. More specifically, we show that there exists a constant $γ> 0$ such that for all $n \ge 0$, $$\ell_R(R/I_{n+1}) - \ell_R(R/I_n) < γn^{d-1}.$$

Linear resolutions of powers and products

The goal of this paper is to present examples of families of homogeneous ideals in the polynomial ring over a field that satisfy the following condition: every product of ideals of the family has a linear free resolution. As we will see, this condition is strongly correlated to good primary decompositions of the products and good homological and arithmetical properties of the associated multi-Rees algebras. The following families will be discussed in detail: polymatroidal ideals, ideals generated by linear forms and Borel fixed ideals of maximal minors. The main tools are Gröbner bases and Sag…

格雷德家族理想 的 resurgence 数

Resurgence number of graded families of ideals

研究格雷德理想家族的 resurgence 和渐近 resurgence 数，探讨其有限性和有理性的条件，以及如何通过 Rees 估值或实际极限计算，并分析积分闭包对渐近 resurgence 的影响。

F-singularities in families

We study the behavior of test ideals and F-singularities in families. In particular, we obtain generic (and non-generic) restriction theorems for test ideals and non-F-pure ideals. Additionally, we study the global behavior of certain canonical linear systems (induced by Frobenius) associated to adjoint line bundles, in families. As a consequence, we obtain some positivity results for pushforwards of some adjoint line bundles and for certain subsheaves of these.

非等维环中的紧闭包

Tight closure in non-equidimensional rings

一个等维局部环是F-有理数的当且仅当一个由参数系生成的理想是紧闭的。一个非等维的局部环是否能有一个由参数系统生成的紧闭理想的问题一直是一个长期存在的公开问题，对于某些类的非等维环，我们证明了这是不可能的。一个关键点是紧闭包在等维环中具有冒号捕获属性，而在非等维环中则没有。我们定义了一个新的闭包运算，它纠正了非等维环中紧闭包的冒号捕获特性的缺失。这种闭包运算在环是等维的情况下与紧封闭一致，并证明了局部环的F-合理性等价于一个参数系关于这种新的闭包运算是封闭的.

线性级数分次族与理想的重数

Multiplicities of graded families of linear series and ideals

本文给出了域k上的射影概形的渐近极限存在的简单的充要条件。当I_n是m_R-准素理想的分次族时，给出了d维局部环R上极限\lim_{i\rightarrow\infty} \frac{\ell_R(R/I_n)}{n^d}存在的充要条件.我们给出了一些应用。

Witt 环上理想的紧闭包

Tight closure of ideals on Witt rings

引入 Witt 环上理想的紧闭包和参数系的拟紧闭性，刻画拟 F-理性，并研究闭包算子与整闭包的关系。

超曲面的$F$ -跳跃理想和$F$ -Jacobian理想

$F$-jumping and $F$-Jacobian ideals for hypersurfaces

为了研究正特征超曲面的奇异性，我们引入了两类理想：$F$ -跳跃理想和$F$ -Jacobian理想.这两个系列都是使用由Blickle、Mustageland和Smith引入的$D$ -模块$M_α$定义的。利用$F$ -跳跃理想与广义检验理想之间的强联系，给出了超曲面的$F$ -跳跃数的一个刻画.此外，我们给出了一个算法，确定某些数字是否是$F$ -跳跃数。另外，我们利用$F$ -Jacobian理想研究了超曲面奇异性的内在性质。特别地，我们给出了$F$ -正则性的条件。此外，$F$ -雅可比理想的行为类似于多项式的雅可比理想。使用开发的技术来研究这两个新的家庭的理想，我们提供了测试理想，广义测试理想，和广义Lyubeznik数的超曲面之间的关系。

渐近多重性

Asymptotic Multiplicities

给出了分次理想族作为极限的重数存在性定理的几个新的应用，包括分次理想族的一个非常一般的Minkowski型不等式，局部体积作为极限的一个非常一般的存在性公式和模的重数作为极限的一个非常一般的存在性定理.

On the functoriality of marked families

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $\mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic…

Free divisors, blowup algebras of Jacobian ideals, and maximal analytic spread

Free divisors form a celebrated class of hypersurfaces which has been extensively studied in the past fifteen years. Our main goal is to introduce four new families of homogeneous free divisors and investigate central aspects of the blowup algebras of their Jacobian ideals. For instance, for all families the Rees algebra and its special fiber are shown to be Cohen-Macaulay -- a desirable feature in blowup algebra theory. Moreover, we raise the problem of when the analytic spread of the Jacobian ideal of a (not necessarily free) polynomial is maximal, and we characterize this property with tool…

Asymptotic Multiplicities of Graded Families of Ideals and Linear Series

We find simple necessary and sufficient conditions on a local ring $R$ of dimension $d$ for the limit $$ \lim_{i\rightarrow\infty}\frac{\ell_R(R/I_n)}{n^d} $$ to exist whenever $\{I_n\}$ is a graded family of $m_R$-primary ideals, and give a number of applications. We also give simple necessary and sufficient conditions on projective schemes over a field $k$ for asymptotic limits of the growth of all graded linear series of a fixed Kodaira-Iitaka dimension to exist.

关于二项边理想的符号$F$-分裂性

On the symbolic $F$-splitness of binomial edge ideals

本文研究了二项边理想族的符号$F$-分裂性及其符号吹胀代数的强$F$-正则性，利用Fedder类准则和相关图的组合性质对上述情形进行了探讨。

理想族的Lojasiewicz指数，Rees混合重数和Newton滤子

Lojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

我们给出了一个广泛的一类n元组的理想$(I_1,..., I_n)$在$Ø_n$使用固定的牛顿过滤所提供的信息的Eschojasiewicz指数的表达式。为了得到这个表达式，我们考虑Rees混合多重性的Rejojasiewicz指数的重新表述。因此，我们得到了一类广义的半加权齐次函数$(\mathbb{C}^n,0)\to (\mathbb{C},0)$，对于这类函数，其梯度映射$\nabla f$的最大可能值为零.

关于Noetherian局部环上分级理想族的乘数

Multiplicities of graded families of ideals on Noetherian local rings

本文研究了Noetherian局部环上分级理想族的乘数概念，推广了传统理想乘数，并证明了多个经典定理在该框架下的泛化，包括混合乘数、Rees定理和Minkowski不等式。

理想的梯度族的渐近正则性

Asymptotic regularity of graded families of ideals

本研究证明齐次理想梯度族渐近正则性在多种条件下存在，给出组合解释，否定若干极限存在性，并构造显式 Gröbner 基。

Daigle & Freudenberg 对希尔伯特第十四问题反例的有限生成理想

The finite generation ideal for Daigle & Freudenberg's counterexample to Hilbert's fourteenth problem

本文计算了 Daigle 和 Freudenberg 对希尔伯特第十四问题反例的有限生成理想，证明其由三个无限族与另一不变式构成的 SAGBI 基生成，并利用该基完成计算。

Properties of analogues of Frobenius powers of ideals

Let $R=\mathbb{K}[X_1, \ldots , X_n ]$ be a polynomial ring over a field $\mathbb{K}$. We introduce an endomorphism $\mathcal{F}^{[m]}: R \rightarrow R $ and denote the image of an ideal $I$ of $R$ via this endomorphism as $I^{[m]}$ and call it to be the $m$ \textit{-th square power} of $I$. In this article, we study some homological invariants of $I^{[m]}$ such as regularity, projective dimension, associated primes and depth for some families of ideals e.g. monomial ideals.

几类分支线性单项理想

Some families of componentwise linear monomial ideals

设R=k[x_1，...，x_n]是域k上的多项式环。令J={j_1，...，j_t}是[n]={1，.，n}，并且令m_J表示理想（x_{j_1}，.，x_{j_t}）。给定子集J_1，...，[n]的J_s和正整数a_1，.，a_s，我们研究理想的形式I= m_{J_1} ^{a_1} \cap. \cap m_{J_s} ^{a_s}。这些理想自然出现，例如，在研究胖点，四面体曲线，和亚历山大对偶的squarefree单项理想。我们的主要焦点是确定这种形式的理想何时是分量线性的。利用多拟阵性，我们证明了当s <= 3或对所有i \neq j，当J_i \cup J_j = [n]时，I总是分量线性的.当s >= 4时，我们给出例子说明I可能是分量线性的，也可能不是.我们将这些结果应用于多投射空间中一般胖点的小集合的理想，并且我们通过计算s=2情况下的分次贝蒂数来扩展Fatabbi，Lorenzini，Valla和第一作者的工作。由于当char（k）=0时，分量线性理想满足Herzog，Huneke和Srinivasan的多重性猜想，我们的工作也产生了这个猜想成立的新情况。

Convex bodies and asymptotic invariants for powers of monomial ideals

Continuing a well established tradition of associating convex bodies to monomial ideals, we initiate a program to construct asymptotic Newton polyhedra from decompositions of monomial ideals. This is achieved by forming a graded family of ideals based on a given decomposition. We term these graded families powers since they generalize the notions of ordinary and symbolic powers. Asymptotic invariants for these graded families are expressed as solutions to linear optimization problems on the respective convex bodies. This allows to establish a lower bound on the Waldschmidt constant of a monomi…

Ratliff-Rush闭包的一种算法

An Algorithm for Computing the Ratliff-Rush Closure

设I \subset K[x，y]是一个<x，y>-准素单项理想，其中K是一个域.本文给出了理想I=<m_0，…，m_{n} >，只要m_{i}包含在理想<x^{ a_{n} }，y^{b_0}>的整闭包中。这概括了Crispin \cite{Cri}的工作。此外，它还对\cite{HJLS}中的一些问题作了推广和回答，并使我们能够构造Ratliff-Rush理想的无限族.

渐进 $\mathrm{v}$-数与格雷德理想族的牛顿-奥科诺夫区域

Asymptotic $\mathrm{v}$-number of graded families of ideals and the Newton-Okounkov region

本文研究了格雷德理想族的渐进 $\mathrm{v}$-数，证明其极限存在并给出计算方法，同时通过牛顿-奥科诺夫区域提供了组合解释，并探讨了渐进准线性性质及 $\mathrm{v}$ 与 $\operatorname{reg}$ 的关系。

仿射或射影概形族的Groebner基

Groebner bases for families of affine or projective schemes

设$I$是交换Noether环$A$上多项式环$A[x]=A[x_1,...,x_n]$的理想.几何上$I$定义了$\Spec(A)$上的一类仿射概型：对于$\p\in\Spec(A)$，$\p$上的纤维是剩余域$k(\p)$上仿射空间的闭子概型，它由$I$在正则映射$σ_\p:A[x]\to k(\p)[x]$下的扩张决定.如果$I$是齐次的，则存在类似的投影设置，但是定义纤维的理想设置还是$\sigI$。对于一个选定的长期秩序，这个理想有一个独特的减少Gröbner基础，这是已知的包含相当多的几何信息的纤维。我们研究的行为，这个基础上，为不同的$\p$和证明存在一个典型的分解的基础空间$\Spec(A)$到1000多个局部闭子集的减少Gröbner基地的纤维可以参数化在一个合适的方式。

The regularity of some families of circulant graphs

We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity.

渐近不变量的渐进族对偶性

Duality for asymptotic invariants of graded families

本文研究了渐进族的对偶性，通过交换子加性和超加性序列并反转其渐近增长常数，揭示了其在代数几何中的应用，包括Macaulay-Matlis对偶性和jet分离序列的对偶关系。

法曲面奇点上m-准素理想的一般元素

General elements of an m-primary ideal on a normal surface singularity

在本文中，我们展示了如何应用Lê D. T.并讨论了法向曲面奇点上的线性曲线族，得到了这方面的新结果。使用的主要概念是一个具体的定义{ \em一般元素}的理想中的局部环的表面。通过使用{ \em约化}，我们明确了这个概念和线性束的一般元素的基本概念之间的联系。这使我们能够证明一般的米尔诺尔数的不变性（分别为。的判别式的多重性），两个铅笔之间产生两个理想具有相同的整闭包（分别。相关的预测）。我们还表明，我们的定理，适用于两种特殊情况下，一方面完成了一个定理Snoussi的极限切超平面，另一方面给出了一个代数$μ$常数定理的线性家庭的平面曲线。

几何顶点分解与联络

Geometric vertex decomposition and liaison

本文建立了几何顶点分解与联络两种框架间的显式联系，证明几何顶点可分解理想通过高度为1的初等G-联络与不定元理想相连，并由此推出多个已知理想族为glicci。

通过 $p$-族视角研究 Cartier 代数

Cartier algebras through the lens of $p$-families

本文通过 $F$-分次理想系统研究 Cartier 代数，证明在 Gorenstein 强 $F$-正则局部环中，对于称为 $p$-族的特殊 $F$-分次系统，强 $F$-正则性与 $F$-分裂等价，并引入 $p$-稳定化操作给出判别准则，最后结合单项式理想给出组合对象以计算 $p$-稳定化。

ε重数及理想族的密度函数

Density functions for epsilon multiplicity and families of ideals

本文为齐次理想的滤过及ε重数构造密度函数，证明其连续性与可微性，并揭示对角子代数重标度重数的连续性。

整环上的星可逆性与$t$有限character

Star-Invertibility and $t$-finite character in Integral Domains

设$A$是一个整环.研究了$A$的整理想族的新条件，得到了$A$具有$t$ -有限特征标（即，$A$的每个非零元素包含在多个$t$ -极大理想中）。我们还研究了与理想的局部可逆性有关的问题。

由图的最小Gröbner基生成的环面理想

Toric ideals of graphs minimally generated by a Gröbner basis

本文研究图环面理想的最小Gröbner基生成问题，证明UMG-图等价于广义鲁棒理想，并证明MG-图类对诱导子图封闭，同时刻画了两类二部MG-图。

Families of Gröbner Degenerations, Grassmannians and Universal Cluster Algebras

Let $V$ be the weighted projective variety defined by a weighted homogeneous ideal $J$ and $C$ a maximal cone in the Gröbner fan of $J$ with $m$ rays. We construct a flat family over $\mathbb A^m$ that assembles the Gröbner degenerations of $V$ associated with all faces of $C$. This is a multi-parameter generalization of the classical one-parameter Gröbner degeneration associated to a weight. We explain how our family can be constructed from Kaveh-Manon's recent work on the classification of toric flat families over toric varieties: it is the pull-back of a toric family defined by a Rees algeb…