Michela Di Marca, Matteo Varbaro
We begin the study of the notion of diameter of an ideal I of a polynomial ring S over a field, an invariant measuring the distance between the minimal primes of I. We provide large classes of Hirsch ideals, i.e. ideals with diameter not larger than the codimension, such as: quadratic radical ideals of codimension at most 4 and such that S/I is Gorenstein, or ideals admitting a square-free complete intersection initial ideal.
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A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set. For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials. We characterize the codimension 2 strongly robust toric ideals by their Gale diagrams. This gives a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.
We provide explicit combinatorial descriptions of the primary components of codimension two lattice basis ideals. As an application, we compute the set of parameters for which a bivariate Horn system of hypergeometric differential equations is holonomic.
Structure theorems for Gorenstein ideals of codimension four with small number of generators
本文用表示论方法研究余维数为四的Gorenstein理想的极小自由消解，引入高阶结构映射族，并证明由六个元素极小生成的该类理想必为余维数为三的Gorenstein理想的超平面截面，同时提出七、八个生成元情形的猜想。
Koszul homology of codimension 3 Gorenstein ideals
本文计算了余维为3的Gorenstein理想的Koszul同调.我们发现过滤的Koszul同调的模块与纯自由分解，并完全描述这些决议。我们还考虑Huneke-Ulrich偏差2理想。
Normal Toric Ideals of Low Codimension
余维为2的正规环面理想都是由具有无平方初始单项式的Grobner基极小生成的。给出了一个多项式时间算法，用于检验一个具有固定余维的复曲面理想是否正规。
Cohen-Macaulay Monomial Ideals of Codimension 2
给出了余维为2的Cohen Macaulay单项理想的一个结构定理，并描述了这类理想的所有可能的关系矩阵。在理想有线性分解的情况下，关系矩阵可以用一个连通弦图的生成树来表示，并且该图的每对极大团至多有一个公共点.
In this paper we define an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the Tor algebra. This family is likely to play a key role in classifying perfect ideals with five generators of type two.
A degree bound for codimension two lattice ideals
Herzog和Srinivasan已经证明了对于任何齐次k-代数，其度是由合合的最大度的函数上有界的。结合Peeva和Sturmfels的合冲四边形分解和一个精细的案例分析，我们证明了这个限制的界限对余维2格理想成立。
Single spot ideals of codimension 3 and long Bourbaki sequences
设K是域，S = K[x1，...，xn]是多项式环。单点理想I =< S是分次理想，其局部上同调H^i_ \mm（S/I），i< dim S/I且\mm =（x1，.，xn），在i =深度S/I处，仅具有非平凡值N，有限长度模。我们认为表征单点理想（长）布尔巴基序列。余维2的情况已经得到了很好的研究。在本文中，我们重点讨论余维3的情况。
We study Gorenstein ideals of codimension $4$ derived from generic doublings of almost complete intersection perfect ideals of codimension $3$. We also investigate spinor coordinates of such Gorenstein ideals with $8$ and $9$ generators. For an ideal $J$ of commutative ring $R$, the $R/J$ module $J/J^2$ is called conormal module and $R/J$-dual of $J/J^2$ is called normal module. We study properties of conormal and normal modules of almost complete intersection perfect ideals of codimension $3$.
One deals with catalectic codimension two perfect ideals and certain degenerations thereof, with a view towards the nature of their symbolic powers. In the spirit of [10] one considers linearly presented such ideals, only now in the situation where the number of variables is sufficiently larger than the size of the matrix, yet still stays within reasonable bounds.
The class of equidimensional polymatroidal ideals are studied. In particular, we show that an unmixed polymatroidal ideal is connected in codimension one if and only if it is Cohen-Macaulay. Especially a matroidal ideal is connected in codimension one precisely when it is a squarefree Veronese ideal. As a consequence we indicate that for polymatroidal ideals, the Serre's condition $(S_n)$ for some $n\geq 2$ is equivalent to Cohen-Macaulay property. We also give a classification of generalized Cohen-Macaulay polymatroidal ideals.
This work is about symbolic powers of codimension two perfect ideals in a standard polynomial ring over a field, where the entries of the corresponding presentation matrix are general linear forms. The main contribution of the present approach is the use of the birational theory underlying the nature of the ideal and the details of a deep interlacing between generators of its symbolic powers and the inversion factors stemming from the inverse map to the birational map defined by the linear system spanned by the generators of this ideal. A full description of the corresponding symbolic Rees alg…
In the present note we study certain arrangements of codimension $2$ flats in projective spaces, we call them "Fermat arrangements". We describe algebraic properties of their defining ideals. In particular, we show that they provide counterexamples to an expected containment relation between ordinary and symbolic powers of homogeneous ideals.
We analyze the structure of spinor coordinates on resolutions of Gorenstein ideals of codimension four. As an application we produce a family of such ideals with seven generators which are not specializations of Kustin-Miller model.
Inverse Grobner Basis Problem in Codimension Two
一般的联系是用来计算一个素理想，使得素理想的初始理想的根等于一个给定的余维2单项式理想的根，该理想具有一个Cohen-Macaulay商环。
We show that the union of certain codimension two flats in Fermat hyperplane arrangements does not have the symbolic cube of the defining ideal contained in the square of the ideal.
Adjoint ideals along closed subvarieties of higher codimension
本文引入了伴随理想层的概念，并利用特征$p$方法研究了伴随理想层的局部性质。当$X$是光滑复簇$A$的正规Gorenstein闭子簇时，我们给出了$A$沿$X$的伴随理想层$\adj_X(A)$的一个限制性质，它涉及到L.C.I. $X$理想层$\mathcal{D}_X$。证明依赖于修改的广义测试理想的原和吉田。
Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective …
Associated primes of powers of closed neighborhood ideals and diameters of graphs
研究闭邻域理想幂的关联素与图直径的关系，证明直径上限为7t-8，并证明该界为紧致。
This article is built upon three main ideas. First, for a class of monomial ideals, it is proven that the multiplicity of an ideal equals the number of realizations of its codimension (an intuitive concept that we define later). Next, for an arbitrary graph G, we construct a monomial ideal M_G, and show that the chromatic number of G is equal to the codimension of M_G. Finally, for a class of graphs, we give a formula that computes the chromatic polynomial of G, evaluated at the chromatic number of G, in terms of the codimension and multiplicity of M_G. In particular, the formula applies to al…
Associated primes of the second power of closed neighborhood ideals of graphs
本文研究极大齐次理想作为闭邻域理想二次幂相伴素理想的简单图，证明此类图直径至多为6，且直径为2的图必为顶点直径-2-临界图。
On the Rees algebra of certain codimension two perfect ideals
设X是P^2中的光滑点集，I = \oplus_{t >= d} I_t是X的定义理想.本文给出了由I_{d+1}生成的理想的Rees代数R（I_{d+1}）的一组定义方程.本文的研究结果完全回答了许多作者所研究的关于P^2沿X中点爆破的投射嵌入的定义理想的问题。本文还研究了由I_t生成的理想的Rees代数R（I_t）在t变大时的渐近性质。
The famous structure theorem of Buchsbaum and Eisenbud gives a complete characterization of Gorenstein ideals of codimension 3 and their minimal free resolutions. We generalize the ideas of Buchsbaum and Eisenbud from Gorenstein ideals to Gorenstein algebras and present a description of Gorenstein algebras of any odd codimension. As an application we study the canonical ring of a numerical Godeaux surface.
The structure of almost Cohen-Macaulay $3$-generated ideals of codimension $2$ in terms of matrix theory
研究通过矩阵理论描述了codimension为2的几乎Cohen-Macaulay 3生成理想结构，基于自由分解和潜在数据，引入了水平矩阵概念，并给出其存在条件。
Cohen--Macaulay ideals of codimension two and the geometry of plane points
本文通过分析 Hilbert–Burch 矩阵研究余维数二 Cohen–Macaulay 理想及平面点几何，并证明超平面排列 Jacobian 理想猜想。
Let $I_X$ be the saturated homogeneous ideal defining a codimension two arithmetically Cohen-Macaulay scheme $X \subseteq \mathbb{P}^n$, and let $I_X^{(m)}$ denote its $m$-th symbolic power. We are interested in when $I_X^{(m)} = I_X^m$. We survey what is known about this problem when $X$ is locally a complete intersection, and in particular, we review the classification of when $I_X^{(m)} = I_X^m$ for all $m \geq 1$. We then discuss how one might weaken these hypotheses, but still obtain equality between the symbolic and ordinary powers. Finally, we show that this classification allows one to…
We focus on the structure of a homogeneous Gorenstein ideal $I$ of codimension three in a standard polynomial ring $R=\kk[x_1,\ldots,x_n]$ over a field $\kk$, assuming that $I$ is generated in a fixed degree $d$. For such an ideal $I$ this degree comes along with the minimal number of generators of $I$ and the degree of the entries of the associated skew-symmetric matrix in a simple formula. We give an elementary characteristic-free argument to the effect that, for any such data linked by this formula, there exists a Gorenstein ideal $I$ of codimension three filling them. We conjecture that, f…
The goal of this paper is the fine structure of the ideals in the title, with emphasis on the properties of the associated Rees algebra and the special fiber. The watershed between the present approach and some of the previous work in the literature is that here one does not assume that the ideals in question satisfy the common generic properties. One exception is a recent work of N. P. H. Lan which inspired the present work. Here we recover and extend his work. We strongly focus on the behavior of the ideals of minors of the corresponding so-called Hilbert--Burch matrix and on conjugation fea…
Every artinian quotient of $K[x,y]$ has the strong Lefschetz property if $K$ is a field of characteristic zero or is an infinite field whose characteristic is greater than the regularity of the quotient. We improve this bound in the case of monomial ideals. Using this we classify when both bounds are sharp. Moreover, we prove that the artinian quotient of a monomial ideal in $K[x,y]$ always has the strong Lefschetz property, regardless of the characteristic of the field, exactly when the ideal is lexsegment. As a consequence we describe a family of non-monomial complete intersections that alwa…
We prove a sharp upper bound for the projective dimension of ideals of height two generated by quadrics in a polynomial ring with arbitrary large number of variables.
The canonical trace of Cohen-Macaulay algebras of codimension 2
本文证明了Herzog关于余维数2的Cohen-Macaulay代数典范迹可由极大子式泛型情形特殊化得到的猜想，并分类了高度为2的近Gorenstein单项式理想。
On the Betti numbers of some Gorenstein ideals
设$R$是域上的多项式环，$I$是余维为$g\ge3$且初度为$p\ge2$的齐次Gorenstein理想.证明了$I$的在度$p$中的极小生成元$ν(I_p)$的个数由$ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$上有界，$ν_0={p+g-1\choose g-1}-{p+g-3\choose g-1}$是余维$g$和初度$p$的极值Gorenstein代数的定义理想的极小生成元个数.此外，如果$ν(I_p)=ν_0$，则$I$本身是极值的。
Stanley decompositions and partitionable simplicial complexes
我们研究了Stanley分解，证明了Stanley关于Stanley分解的猜想蕴含着他关于可分Cohen-Macaulay单纯复形的猜想.我们还证明了这些定理的所有Cohen-Macaulay单项理想的余维2和所有Gorenstein单项理想的余维3。
Let $(S, \mathfrak n) $ be a regular local ring and let $I \subseteq \mathfrak n^2 $ be a perfect ideal of $S. $ Sharp upper bounds on the minimal number of generators of $I$ are known in terms of the Hilbert function of $R=S/I. $ Starting from information on the ideal $I, $ for instance the minimal number of generators, a difficult task is to find good bounds on the minimal number of generators of the leading ideal $I^* $, which defines the tangent cone of $R$ or to give information on its graded structure. Motivated by papers of S.C. Kothari, S. Goto ${\it{et al.}}$ concerning the leading id…
I describe the projective resolution of a codimension 4 Gorenstein ideal, aiming to extend Buchsbaum and Eisenbud's famous result in codimension 3. The main result is a structure theorem stating that the ideal is determined by its (k+1) x 2k matrix of first syzygies, viewed as a morphism from the ambient regular space to the Spin-Hom variety SpH_k in Mat(k+1,2k). This is a general result encapsulating some theoretical aspects of the problem, but, as it stands, is still some way from tractable applications.
Depth of Binomial Edge Ideals in terms of Diameter and Vertex Connectivity
本文基于图的直径和顶点连通度等不变量，对满足深度上下界关系$L(G)+1=U(G)$的简单连通非完全图进行了结构分类，并计算了此类图二项边理想$S/J_G$的深度。
Codimension Theorems for Complete Toric Varieties
设X是具有齐次坐标环S的完备复曲面簇。本文计算了由dim（X）+1个X上不同时为零的齐次多项式生成的S的理想的临界度的余维数的上、下界.
On a classification of ideals of local rings for irreducible curve singularities
考虑了当环是具有不可约曲线奇点的局部环时理想的余维数分类问题。在本文中，我们介绍了一个系统的方法来解决这个问题。
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…
We study how to lift Markov bases and Gröbner bases along linear maps of lattices. We give a lifting algorithm that allows to compute such bases iteratively provided a certain associated semigroup is normal. Our main application is the toric fiber product of toric ideals, where lifting gives Markov bases of the factor ideals that satisfy the compatible projection property. We illustrate the technique by computing Markov bases of various infinite families of hierarchical models. The methodology also implies new finiteness results for iterated toric fiber products.
Evolutions, Symbolic Squares, and Fitting Ideals
给定环或域$k$上的约化局部代数$T$，研究是否存在非平凡代数满射$R\to T$诱导同构$Ω_{R/k}\otimes T \to Ω_{T/k}$ .这样的映射，称为演化，在赫克代数的研究中自然出现，就像怀尔斯、泰勒-怀尔斯和弗拉克最近的工作中隐含的那样。我们证明了代数$T$的非平凡演化的存在性可以用定义$T$的理想的符号平方来刻画。我们用Fitting理想刻画了符号正方形。使用这种和其他技术，我们表明，某些类的减少代数-余维2科恩-麦考利，余维3 Gorenstein，licci代数一般，和其他一些-承认没有非平凡的演变。另一方面，我们给出了余维为3的约化Cohen-Macaulay代数的非平凡演化存在于每个正特征中的例子。
Betti strata of height two ideals
本文确定了二元多项式环R的分次Artin行列式A的参数化族G（H）的Betti层的余维数. Betti层G（B，H）参数化了所有具有由关系度B确定的分次贝蒂数的参数。我们的方法是将G（B，H）确定为行列式簇的乘积。然后我们使用M的一个结果。Boij，它减少了余维的计算，以表明最特殊的层具有预期的余维。我们还表明，Betti地层的封闭是Cohen-Macaulay，和较低的地层的联合。作为应用，我们确定了A的可能的Hilbert函数，给定的socle类型;我们确定了R的t个一般的足够水平代数子的交集的Hilbert函数，每个子都是给定的Hilbert函数。
Generic Initial Ideals And Graded Artinian Level Algebras Not Having The Weak-Lefschetz Property
我们找到了一个充分条件$\H$是不水平的基础上减少数。特别地，我们证明了具有Hilbert函数$\H=(h_0,h_1,..., h_{d-1}>h_d=h_{d+1})$的余维为3的分次Artin代数如果$h_d\le 2d+3$不可能是水平的，并且存在$\H$型的余维为3的水平O-序列for $h_d \ge 2d+k$ for $k\ge 4$ .进一步证明了当$β_{1,d+2}(I^{\rm lex})=β_{2,d+2}(I^{\rm lex})$为余维时，$\H$不是水平的;当$β_{1,d+2}(\Gin(I))=β_{2,d+2}(\Gin(I))$为余维时，任何余维3的Artin分次代数$A=R/I$都不是水平的.在这种情况下，$A$的希尔伯特函数不必满足条件$h_{d-1}>h_d=h_{d+1}$。此外，我们还证明了具有Weak-Lefschetz性质的每一个余维$n$分次Artin水平代数都有严格单峰Hilbert函数，对于每一个$d > θ$，它在$(h_{d-1}-h_{d}) \le (n-1)(h_d-h_{d+1})$上有一个增长条件，其中$$ h_0<h_1<...<h_α=...=h_θ>...>h_{s-1}>h_s. $$特别地，我们发现如果$A$是余维3的，那么对于每一个$θ< d <s$和$h_{s-1}\le 3 h_s$，$(h_…
Let $R$ be a Noetherian commutative ring of positive dimension. The Hochster-Huneke graph of $R$ (sometimes called the dual graph of Spec $R$ and denoted by $\mathcal{G} (R)$) is defined as follows: the vertices are the minimal prime ideals of $R$, and the edges are the pairs of prime ideals $(P_1,P_2)$ with height $(P_1 + P_2) = 1$. If $R$ satisfies Serre's property $(S_2)$, then $\mathcal{G} (R)$ is connected. In this note, we provide lower and upper bounds for the maximum diameter of Hochster-Huneke graphs of Stanley-Reisner rings satisfying $(S_2)$. These bounds depend on the number of var…
We describe all Mathieu-Zhao spaces of $k[x_1,\cdots,x_n]$ ($k$ is an algebraically closed field of characteristic zero) which contains an ideal of finite codimension. Furthermore we give an algorithm to decide if a subspace of the form $I+kv_1+\cdots+kv_r$ is a Mathieu-Zhao space, in case the ideal $I$ has finite codimension.
A projectively normal Calabi-Yau threefold $X \subseteq \mathbb{P}^n$ has an ideal $I_X$ which is arithmetically Gorenstein, of Castelnuovo-Mumford regularity four. Such ideals have been intensively studied when $I_X$ is a complete intersection, as well as in the case where $X$ is codimension three. In the latter case, the Buchsbaum-Eisenbud theorem shows that $I_X$ is given by the Pfaffians of a skew-symmetric matrix. A number of recent papers study the situation when $I_X$ has codimension four. We prove there are 16 possible betti tables for an arithmetically Gorenstein ideal $I$ with $\math…
We study the Rees algebra of a perfect Gorenstein ideal of codimension 3 in a hypersurface ring. We provide a minimal generating set of the defining ideal of these rings by introducing a modified Jacobian dual and applying a recursive algorithm. Once the defining equations are known, we explore properties of these Rees algebras such as Cohen-Macaulayness and Castelnuovo-Mumford regularity.
A minor is principal means it is defined by the same row and column indices. Let $X$ be a square generic matrix, $K[X]$ the polynomial ring in entries of $X$, over an algebraically closed field, $K$. For fixed $t\leq n$, let $\mathfrak P_t$ denote the ideal generated by the size $t$ principal minors of $X$. When $t=2$ the resulting quotient ring $K[X]/\mathfrak P_2$ is a normal complete intersection domain. When $t>2$ we break the problem into cases depending on a fixed rank, $r$, of $X$. We show when $r=n$ for any $t$, the respective images of $\mathfrak P_t$ and $\mathfrak P_{n-t}$ in the lo…
Depths of Powers of the Edge Ideal of a Tree
当I是树或森林的边理想时，给出了R/I^t深度的下界，其中t至少为一个。边界是根据树的直径给出的，或者在森林的情况下，连接组件的最大直径和连接组件的数量。这些下界提供了深度稳定的功率的下界。
Generic models of licci ideals parametrized by Schur functors
本文利用与 Schur 函子相关的分区对参数化余维 3 的 licci 理想的 Herzog 类，并组合描述其链接图，进而得到 Tor 代数乘法分类及结构定理。
The acquisition of the defining equations of Rees algebras is a natural way to study these algebras and allows certain invariants and properties to be deduced. In this paper, we consider Rees algebras of codimension 2 perfect ideals of hypersurface rings and produce a minimal generating set for their defining ideals. Then, using generic Bourbaki ideals, we study Rees algebras of modules with projective dimension one over hypersurface rings. We describe the defining ideal of such algebras and determine Cohen-Macaulayness and other invariants.
F-singularities of pairs and Inversion of Adjunction of arbitrary codimension
本文将F-正则环和F-纯环的概念推广到环$R$和理想$\a \subset R$的对$(R,\a^t)$，并研究了这些性质。这些“F-奇点对”对应于奇点对任意余维在双有理几何。通过这种对应，我们证明了任意余维伴随的逆，即对于光滑簇$X$和闭子概型$Y \subsetneq X$的一对$(X,Y)$，如果对正规$\Q$闭子簇$Z \subsetneq X$的限制$(Z, Y|_Z)$是klt（resp. lc），则对$(X,Y+Z)$为plt（resp. lc）在$Z$附近。
Let $G$ be a graph and let $I$ be the edge ideal of $G$. Our main results in this article provide lower bounds for the depth of the first three powers of $I$ in terms of the diameter of $G$. More precisely, we show that $\depth R/I^t \geq \left\lceil{\frac{d-4t+5}{3}} \right\rceil +p-1$, where $d$ is the diameter of $G$, $p$ is the number of connected components of $G$ and $1 \leq t \leq 3$. For general powers of edge ideals we show
For which monomial supports do most polynomials generate a prime ideal? We give necessary and sufficient conditions for the radical of the ideal to be prime over an algebraically closed field. In characteristic zero, the same conditions give primeness. As an application we show that under the same combinatorial conditions on Newton polytopes, the stable intersection of tropical hypersurfaces is connected through codimension one.
Complete intersections in toric ideals
我们给出的例子表明，在大于1的维数或大于2的余维中，存在环面理想I_A，使得包含在I_A中的相同维数的二项式理想都不是完全交.这一结果在稀疏消去理论和Horn型偏微分方程组的研究中具有重要意义。
In this paper we consider the problem of finding explicitly canonical ideals of one-dimensional Cohen-Macaulay local rings. We show that Gorenstein ideals contained in a high power of the maximal ideal are canonical ideals. In the codimension two case, from a Hilbert-Burch resolution, we show how to construct canonical ideals of curve singularities. Finally, we translate the problem of the analytic classification of curve singularities to the classification of local Artin Gorenstein rings with suitable length.
On Rees algebras of linearly presented ideals and modules
本文在更弱的局部生成元数目条件下，确定了线性呈现的完美高度二理想的 Rees 代数的定义理想，并利用 Bourbaki 理想将结果推广到投影维数为一的线性呈现模。
If I is an ideal in a Gorenstein ring S and S/I is Cohen-Macaulay, then the same is true for any linked ideal I'. However, such statements hold for residual intersections of higher codimension only under very restrictive hypotheses, not satisfied even by ideals as simple as the ideal L_n of minors of a generic 2 x n matrix when n>3. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I. For example, we prove that if K is the residua…
L. Szpiro's conjecture on Gorenstein algebras in codimension 2
余维为2的Gorenstein A-代数R是完全有限A-代数，使得R=Ext^2（R，A）作为R-模成立，A是Cohen-Macaulay局部环，dim（A）-dim_A（R）=2.证明了这类代数的一个结构定理，改进了M.格拉西特别注意的问题是如何环结构的R编码在其希尔伯特决议。本文证明了当对由R在A上的表示矩阵导出的行列式理想施加弱深度条件时，R自动成为环。讨论了Gorenstein代数与M.格拉西澄清。P^4中的正则曲面的适用性问题在这些研究中起到了指导作用。
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