Lê Tuân Hoa, Eero Hyry
We give two kinds of bounds for the Castelnuovo-Mumford regularity of the canonical module and the deficiency modules of a ring, respectively in terms of the homological degree and the Castelnuovo-Mumford regularity of the original ring.
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This paper gives explicit formulas for the reduction number and the Castelnuovo-Mumford regularity of projective monomial curves.
Bounds on the Castelnuovo-Mumford regularity of the associated graded modules of k-Buchsbaum modules M are given in terms of k and some other invariants of M.
Asymptotic linear bounds for the Castelnuovo-Mumford regularity
我们证明的Castelnuovo-Mumford正则性的某些过滤齐次理想的Rees代数不需要Noether的渐近线性界。
New upper and lower bounds on the Castelnuovo-Mumford regularity are given in terms of the Hilbert coefficients. Examples are provided to show that these bounds are in some sense nearly sharp.
In this paper we give a nice formula for the Castelnuovo-Mumford regularity of the Segre product of modules, under some suitable hypotheses. This extends recent results of David A. Cox, and Evgeny Materov (2009).
An upper bound for the Castelnuovo-Mumford regularity of the associated graded module of an one-dimension module is given in term of its Hilbert coeffcients. It is also investigated when the bound is attained.
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.
We show that the Eisenbud-Goto conjecture holds for (homogeneous) seminormal simplicial affine semigroup rings. Moreover, we prove an upper bound for the Castelnuovo-Mumford regularity in terms of the dimension, which is similar as in the normal case. Finally, we compute explicitly the regularity of full Veronese rings.
In this article, we study the Castelnuovo-Mumford regularity and Gorenstein properties of the fiber cone. We obtain upper bounds for the Castelnuovo-Mumford regularity of the fiber cone and obtain sufficient conditions for the regularity of the fiber cone to be equal to that of the Rees algebra. We obtain a formula for the canonical module of the fiber cone and use it to study the Gorenstein property of the fiber cone.
We present algorithms for the computation of the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring of equigenerated $\mm$-primary ideals in two variables. Applying these algorithms, we find a counter-example to a conjecture of Eisenbud and Ulrich which states that these regularities are equal.
These notes are an introduction to some basic aspects of the Castelnuovo-Mumford regularity and related topics such as weak regularity, a*-invariant and partial regularities.
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.
We estimate the Castelnuovo-Mumford regularity of ideals in a polynomial ring over a field by studying the regularity of certain modules generated in degree zero and with linear relations. In dimension one, this process gives a new type of upper bounds. By means of recursive techniques this also produces new upper bounds for ideals in any dimension.
In a recent preprint, Ilse Fischer and Martina Kubitzke, proved the bilinearity of the Segre transform under some restricted hypothesis, motivated by their results we show in this paper the bilinearity of the Segre transform in general. We apply these results to compute the postulation number of a series. Our second application is motivated by the paper of David A. Cox, and Evgeny Materov (2009), where is computed the Castelnuovo-Mumford regularity of the Segre Veronese embedding, we can extend partially their result and compute the Castelnuovo-Mumford regularity of the Segre product of Cohen-…
We study the asymptotic behavior of the Castelnuovo-Mumford regularity along chains of graded ideals in increasingly larger polynomial rings that are invariant under the action of symmetric groups. A linear upper bound for the regularity of such ideals is established. We conjecture that their regularity grows eventually precisely linearly. We establish this conjecture in several cases, most notably when the ideals are Artinian or squarefree monomial.
Projective dimension and Castelnuovo-Mumford regularity of t-spread ideals
本文借助分次分解研究了$t$-分散理想的投射维数与Castelnuovo-Mumford正则度等代数不变量，给出了其上界并识别出使该界限最优的特殊理想类。
Castelnuovo-Mumford regularity and reduction number of smooth monomial curves
我们比较，光滑单项射影曲线，卡斯特-诺沃-芒福德正则性和减少数，我们提出了一个例子，这两个数字不同。然而，我们证明了它们对某类单项曲线是coin-cide的.此外，对于光滑的单项曲线，我们证明了一个不等式，这是强于一个从艾森巴德-后藤猜想。
Let ${\mathfrak p}$ be the defining ideal of the monomial curve ${\mathcal C}(2q+1, 2q+1+m, 2q+1+2m)$ in the affine space ${\mathbb A}_k^3$ parameterized by $(x^{2q +1}, x^{2q +1 + m}, x^{2q +1 +2 m})$ where $gcd( 2q+1,m)=1$. In this paper we compute the resurgence of ${\mathfrak p}$, the Waldschmidt constant of ${\mathfrak p}$ and the Castelnuovo-Mumford regularity of the symbolic powers of ${\mathfrak p}$.
We extend to one dimensional quotients the result of A. Conca and S. Murai on the convexity of the regularity of Koszul cycles. By providing a relation between the regularity of Koszul cycles and Koszul homologies we prove a sharp regularity bound for the Koszul homologies of a homogeneous ideal in a polynomial ring under the same conditions.
On the asymptotic linearity of Castelnuovo-Mumford regularity
设R是单位交换诺特环上的标准分次环，I是R的分次理想.设M是M生成的分次R-模。证明了存在整数e和ρ_M（I）使得对所有大的n，reg（I^nM）= ρ_M（I）n+e.
Castelnuovo-Mumford regularity of products of ideals
我们讨论了Castelnuovo-Mumford正则性在理想和模上的某些运算下的行为，如乘积或幂。特别地，我们证明了即使I是线性型的理想且M是具有线性分解的模，reg（IM）也可以大于reg（M）+reg（I）.另一方面，我们证明了任何线性型的理想的理想的积都有线性分解。我们还讨论了多拟阵理想的情况下，并表明，任何产品的一般汉克尔矩阵的行列式理想有一个线性决议。
This note has two goals. The first is to give a short and self contained introduction to the Castelnuovo-Mumford regularity for standard graded ring $R$ over a general base ring. The second is to present a simple and concise proof of a classical result due to Cutkosky, Herzog and Trung and, independently, to Kodiyalam asserting that the regularity of powers of an homogeneous ideal $I$ of $R$ is eventually a linear function in $v$. Finally we show how the flexibility of the definition of the Castelnuovo-Mumford regularity over general base rings can be used to give a simple characterization of …
Weighted Castelnuovo-Mumford Regularity and Weighted Global GenerationWe introduce and study a notion of Castelnuovo-Mumford regularity suitable for weighted projective spaces.
Evaluations of initial ideals and Castelnuovo-Mumford regularity
本文通过对初始理想关于逆字典序的赋值刻画了Castelnuovo-Mumford正则性。
The purpose of this paper is to give a simple geometric construction of ideals whose Castelnuovo-Mumford regularity is large compared to the generating degree. Moreover, our ideals have the property that the Castelnuovo-Mumford regularity is revealed as late in the resolution as desired.
Castelnuovo-Mumford regularity: Examples of curves and surfaces
Castelnuovo-Mumford正则性在“几何”变换下的行为还没有很好的理解。在本文中，我们关注的例子，这将揭示一些关于这种行为的某些问题。
Let $A$ be a Noetherian standard $\mathbb{N}$-graded algebra over an Artinian local ring $A_0$. Let $I_1,\ldots,I_t$ be homogeneous ideals of $A$ and $M$ a finitely generated $\mathbb{N}$-graded $A$-module. We prove that there exist two integers $k$ and $k'$ such that \[ \mathrm{reg}(I_1^{n_1} \cdots I_t^{n_t} M) \leq (n_1 + \cdots + n_t) k + k' \quad\mbox{for all }~n_1,\ldots,n_t \in \mathbb{N}. \]
Uniform bounds on projective dimension and Castelnuovo-Mumford regularity
本文获得关于标准分次多项式环中齐次理想射影维数和卡斯泰洛尼-穆姆福德正则性的统一有效上界，该上界与变量数无关，扩展了麦卡洛格2012年的关于正则性与syzygies关系的结论。
Bigraded Castelnuovo-Mumford regularity and Gröbner bases
该研究拓展单分次情形下的经典结论，引入双齐次Gröbner基极小生成元双次数的有界区域，揭示其与双分次Castelnuovo-Mumford正则性的关联，为相关计算复杂度分析提供基础。
We present new combinatorial insights into the calculation of (Castelnuovo-Mumford) regularity of graphs.
Matchings in hypergraphs and Castelnuovo-Mumford regularity
该研究将图的匹配数与诱导匹配数等组合不变量推广到超图，并据此为特定超图的 Stanley-Reisner 环正则度建立上界。
We prove that for every integer $k\geq 1$, there exists a connected graph $H_k$ such that $v(H_k)=reg(H_k)+k$, where $v(G)$ and $reg(G)$ denote the $v$-number and the (Castelnuovo-Mumford) regularity of a graph $G$ respectively.
Castelnuovo-Mumford regularity of Ext modules and homological degree
给出了域上多项式环上Ext模的Castelnuovo-Mumford正则性的界，给出了它们的初始度、Castelnuovo-Mumford阶数和两个分次模模的生成元个数.在第二个模是环的情况下，这些一般的界限被细化。其他估计，例如，这些模块的分级件的大小，给出。我们还得到了一个绑定的同调度的Castelnuovo-Mumford正则性。这肯定地回答了Vasconcelos提出的一个问题。
Induced matching, ordered matching and Castelnuovo-Mumford regularity of bipartite graphs
本文刻画了诱导匹配数等于有序匹配数的二部图，并给出其边理想幂的正则度与覆盖理想幂的深度公式，以及K_{m,n}中满足条件子图的计数公式。
We prove new results on the connections between reduction numbers and the Castelnuovo-Mumford regularity of blowup algebras and blowup modules, the key basic tool being the operation of Ratliff-Rush closure. First, we answer in two particular cases a question of M. E. Rossi, D. T. Trung, and N. V. Trung about Rees algebras of ideals in two-dimensional Buchsbaum local rings, and we even ask whether one of such situations always holds. In another theorem we generalize a result of A. Mafi on ideals in two-dimensional Cohen-Macaulay local rings, by extending it to arbitrary dimension (and allowing…
The Eisenbud--Goto conjecture states that $\operatorname{reg} X\le\operatorname{deg} X -\operatorname{codim} X+1$ for a nondegenerate irreducible projective variety $X$ over an algebraically closed field. While this conjecture is known to be false in general, it has been proven in several special cases, including when $X$ is a projective toric variety of codimension $2$. We classify the projective toric varieties of codimension $2$ having maximal regularity, that is, for which equality holds in the Eisenbud--Goto bound. We also give combinatorial characterizations of the arithmetically Cohen--…
We give the Castelnuovo-Mumford regularity of arrangements of (n-2)-planes in P^n whose incidence graph is a sufficiently large complete bipartite graph, and determine when such arrangements are arithmetically Cohen-Macaulay.
In this note, we give a bound for the Castelnuovo-Mumford regularity of a homogeneous ideal $I$ in terms of the degrees of its generators. We assume that $I$ defines a local complete intersection with log canonical singularities.
Buchsbaumness, Macaulayfication and Castelnuovo-Mumford regularity of monomial curves
本文构造了无限类非光滑非Cohen-Macaulay的k-Buchsbaum射影单项式曲线，给出其Macaulay化的生成元，并探讨正则度与k-Buchsbaum性的联系。
We establish strong relationships between the Castelnuovo-Mumford regularity and the Ratliff-Rush closure of an ideal. Our results have several interesting consequences on the computation of the Ratliff-Rush closure, the stability of the Ratliff-Rush filtration, the invariance of the reduction number, and the computation of the Castelnuovo-Mumford regularity of the Rees algebra and the fiber ring. In particular, we prove that the Castelnuovo-Mumford regularity of the Rees algebra and of the fiber ring are equal for large classes of monomial ideals in two variables, thereby verifying a conjectu…
Castelnuovo-Mumford regularity and extended degree
本文的主要结果表明，局部环的切锥的Castelnuovo-Mumford正则性是由维数和任意扩张度有效有界的.由此得出，对于给定维数和扩张度的局部环，只有有限个Hilbert-Samuel函数.
We prove a sharp inequality relating the Castelnuovo--Mumford regularity of a coherent ideal sheaf to its log-canonical threshold.
Projective monomial curves correspond to rings generated by monomials of the same degree in two variables. Such rings always have finite Macaulayfication. We show how to characterize the Buchsbaumness and the Castelnuovo-Mumford regularity of these rings by means of their finite Macaulayfication, and we use this method to study the Buchsbaumness and to estimate the Castelnuovo-Mumford regularity of large classes of non-smooth monomial curves in terms of the given monomials.
Bounds for the Castelnuovo-Mumford regularity of modules
本文利用生成元的度和它们之间的关系的度，建立了一个生成分次模的Castelnuovo-Mumford正则性及其对称幂的界.本文将Galligo，Guisti，Caviglia和Sbarra建立的多项式环中齐次理想的界推广到模上，并改进了已有的结果.
Castelnuovo-Mumford Regularity in Biprojective Spaces
定义了双分次模和双分次多项式环的正则性概念。在这种情况下，我们证明了类似的一些经典结果的$m$ -正则性的分次模多项式代数。
Castelnuovo-Mumford Regularity over Scrolls and Splitting Criteria
该研究提出适用于由射影空间上线丛和的射影化得到的scrolls的Castelnuovo-Mumford正则性概念，证明其为经典正则性的自然推广，并建立了向量丛的Horrocks型分裂准则。
We show that the co-chordal cover number of a graph G gives an upper bound for the Castelnuovo-Mumford regularity of the associated edge ideal. Several known combinatorial upper bounds of regularity for edge ideals are then easy consequences of covering results from graph theory, and we derive new upper bounds by looking at additional covering results.
We prove a conjectured upper bound for the Castelnuovo-Mumford regularity of binomial edge ideals of graphs, due to Matsuda and Murai. Indeed, we prove that $\mathrm{reg}(J_G)\leq n-1$ for any graph $G$ with $n$ vertices, which is not a path. Moreover, we study the behavior of the regularity of binomial edge ideals under the join product of graphs.
Finiteness of Hilbert functions and bounds for Castelnuovo-Mumford regularity of initial ideals
Castelnuovo-Mumford正则性和Hilbert系数的界是根据算术度（如果环是约化的）或定义度给出的。由此可以得出，在具有给定算术次数和维数的代数闭域上，只存在有限个与约化代数相关联的希尔伯特函数。一个很好的界也给出了Castelnuovo-Mumford正则性的初始理想，既不依赖于长期的订单，也不对坐标，并保持任何领域。
On the Castelnuovo-Mumford regularity of curve arrangements
该研究探讨复射影平面一般曲线构形的雅可比代数及分次导子模的Castelnuovo-Mumford正则性，提出无需拟齐次性假设的加-删型结果，拓展了相关经典结论的适用范围。
Liaison and Castelnuovo-Mumford regularity
在这篇文章中，我们建立的Castelnuovo-Mumford正则性的射影概形的定义方程的次数的界限。在我们的证明中主要的新成分是证明完全相交有理奇点的一般剩余交点再次具有有理奇点。当适用于理论的剩余交叉点这一圈的想法也揭示了新的光对一些已知的类自由分解的剩余理想。
Castelnuovo-Mumford Regularity and Combinatorial Invariants of Trees
本文研究了树的边理想及其多须变体的Castelnuovo-Mumford正则性，用阶、直径和悬挂顶点数给出组合界，并导出了诱导匹配数和独立数的不等式。
Matrix Schubert varieties are affine varieties arising in the Schubert calculus of the complete flag variety. We give a formula for the Castelnuovo-Mumford regularity of matrix Schubert varieties, answering a question of Jenna Rajchgot. We follow her proposed strategy of studying the highest-degree homogeneous parts of Grothendieck polynomials, which we call Castelnuovo-Mumford polynomials. In addition to the regularity formula, we obtain formulas for the degrees of all Castelnuovo-Mumford polynomials and for their leading terms, as well as a complete description of when two Castelnuovo-Mumfor…
On the behavior of Castelnuovo-Mumford regularity with respect to some functors
我们研究了Castelnuovo-Mumford正则性关于一些经典函子的行为：Tor，正特征的Frobenius函子，取幂或乘积（理想上）。这些推广和完善以前的结果，这些问题上的几个作者。作为一个应用，我们提供的结果的交集的射影概形的子计划的规则性，在适当的几何假设。多个Tor模块的刚性和其消失的表征的结果给出，几何应用的动机。
Irrational Asymptotic Behaviour of Castelnuovo-Mumford Regularity
我们给出了射影3空间中的非奇异曲线的例子，使得它们的理想层的幂的正则性是高度非线性的。这与多项式环中的理想I的情况不同，其中I^n的正则性是n中的线性多项式，n较大。
Castelnuovo-Mumford regularity of unprojections and the Eisenbud-Goto regularity conjecture
针对Eisenbud-Goto正则性猜想，本文提出一种基于非投影过程的反例构造方法，可构造任意固定维数≥3、余维数≥2的反例，同时揭示了奇点对Castelnuovo-Mumford正则性的影响。
We prove the regularity conjecture, namely Eisenbud-Goto conjecture, for a normal surface with rational, Gorenstein elliptic and log canonical singularities. Along the way, we bound the regularity for a dimension zero scheme by its Loewy length and for a curve allowing embedded or isolated point components by its arithmetic degree.
On the Castelnuovo-Mumford regularity of ideals, in dimension 2
当A/I的维数最多为2时，我们给出了多项式环A中齐次理想I的Castelnuovo-Mumford正则性的一个界，该界是根据变量数和生成元的次数给出的。这个界限改进了Caviglia和Sbarra在这种情况下得到的界限。在继续的例子所构建的克莱尔D '克鲁兹和第一作者，我们使用家庭的单项曲线，以建设齐次理想表明，这些界限是相当尖锐的。
A sharp bound for the Castelnuovo-Mumford regularity of subspace arrangements
证明了射影空间的d个线性子空间的排列理想在Castelnuovo和Mumford意义下是d-正则的，回答了B.斯特姆费尔斯特别地，这意味着d个子空间的布置的理想以小于或等于d的度数生成。
Characteristic-free bounds for the Castelnuovo-Mumford regularity
本文研究了多项式环中齐次理想的Castelnuovo-Mumford正则性在变元个数和生成元次数方面的界.特别是我们的目的是给一个积极的回答拜耳和芒福德提出的问题，通过显示，已知的上界的特征零也适用于积极的特点。我们首先分析了Giusti的证明，它提供了特征0的结果，在这种情况下所需的组合属性提供了一些见解。对于一般情况，我们提供了一个新的论点，采用拜耳和斯蒂尔曼标准检测的规律性。
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