On the symmetric and Rees algebras of certain determinantal ideals

On Rees algebras of ideals and modules over hypersurface rings

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.

The equations of Rees algebras of height three Gorenstein ideals in hypersurface rings

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.

Rees algebra and special fiber ring of binomial edge ideals of closed graphs

In this article, we compute the regularity of Rees algebra of binomial edge ideals of closed graphs. We obtain a lower bound for the regularity of Rees algebra of binomial edge ideals. We also study some algebraic properties of the Rees algebra and special fiber ring of binomial edge ideals of closed graphs via algebraic properties of their initial algebra and Sagbi basis theory. We obtain an upper bound for the regularity of the special fiber ring of binomial edge ideals of closed graphs.

Gorensteinness in Rees algebras of powers of parameter ideals

This paper gives a necessary and sufficient condition for Gorensteinness in Rees algebras of the $d$-th power of parameter ideals in certain Noetherian local rings of dimension $d\ge 2$. The main result of this paper produces many Gorenstein Rees algebras over non-Cohen-Macaulay local rings. For example, the Rees algebra $\mathcal{R}(\mathfrak{q}^d)=\oplus_{i\ge 0}\mathfrak{q}^{di}$ is Gorenstein for every parameter ideal $\mathfrak{q}$ that is a reduction of the maximal ideal in a $d$-dimensional Buchsbaum local ring of depth 1 and multiplicity 2.

Rees algebras of ideals of star configurations

In this article we study the defining ideal of Rees algebras of ideals of star configurations. We characterize when these ideals are of linear type and provide sufficient conditions for them to be of fiber type. In the case of star configurations of height two, we give a full description of the defining ideal of the Rees algebra, by explicitly identifying a minimal generating set.

Rees's theorem for arbitrary ideals

This paper has been withdrawn by the authors, due a crucial error in the proof of the main theorem.

Finitely generated symbolic Rees rings of ideals defining certain finite sets of points in P^2

The purpose of this paper is to prove that the symbolic Rees rings of ideals defining certain finite sets of points in the projective plane over an algebraically closed field are finitely generated using a ring theoretical criterion which is known as Huneke's criterion.

Matching numbers and the regularity of the Rees algebra of an edge ideal

The regularity of the Rees ring of the edge ideal of a finite simple graph is studied. We show that the matching number is a lower and matching number~$+1$ is an upper bound of the regularity, if the Rees algebra is normal. In general the induced matching number is a lower bound for the regularity, which can be shown by applying the squarefree divisor complex.

Symbolic Rees algebras and set-theoretic complete intersections

In this paper we extend a result of Cowsik on set-theoretic complete intersection and a result Huneke, Morales and Goto and Nishida about Noetherian symbolic Rees algebras of ideals. As applications, we show that the symbolic Rees algebras of the following ideals are Noetherian and the ideals are set-theoretic complete intersections: (a) the edge ideal of a complete graph, (b) the Fermat ideal and (c) the Jacobian ideal of a certain hyperplane arrangement.

Canonical modules of Rees algebras

We compute the canonical class of certain Rees algebras. Our formula generalizes that of Herzog and Vasconcelos. Its proof relies on the fact that the formation of the canonical module commutes with subintersections in important cases. As an application we treat the classical determinantal ideals and the corresponding algebras of minors.

Asymptotic invariants of ideals with Noetherian symbolic Rees algebra and applications to cover ideals

Let $I$ be an ideal whose symbolic Rees algebra is Noetherian. For $m \geq 1$, the $m$-th symbolic defect, sdefect$(I,m)$, of $I$ is defined to be the minimal number of generators of the module $\frac{I^{(m)}}{I^m}$. We prove that sdefect$(I,m)$ is eventually quasi-polynomial as a function in $m$. We compute the symbolic defect explicitly for certain monomial ideals arising from graphs, termed cover ideals. We go on to give a formula for the Waldschmidt constant, an asymptotic invariant measuring the growth of the degrees of generators of symbolic powers, for ideals whose symbolic Rees algebra…

Minimal generators of the defining ideal of the Rees Algebra associated to a rational plane parameterization with μ=2

We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated to the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

Linear resolutions and Gröbner basis of Hankel determinantal ideals

In this paper, we study the family of determinantal ideals of "close" cuts of Hankel matrices, say $ \mathcal{f} $. We show that the multi-Rees algebra of ideals in $ \mathcal{f} $ is defined by a quadratic Gröbner basis, it is Koszul, normal Cohen-Macaulay domain and it has a nice Sagbi basis. As a consequence of Koszulness, we prove that every product $ I_1\ldots I_l $ of ideals of $ \mathcal{f} $ has linear resolution. Moreover, we show that natural generators of every product $ I_1\ldots I_l $ form a Gröbner basis.

关于圈的$t$-路理想的里斯代数

On the Rees algebras of $t$-path ideals of cycles

本文研究$n$-圈的$t$-路理想$I_t(C_n)$的里斯代数性质，重点分析该理想为纤维型的情形。

Multi-Rees algebras on principal ideal rings

When $R$ is a Noetherian ring and we have a family of ideals in which every ideal contains at least one nonzero divisor, then it is already known that the defining ideal of the multi-Rees algebra of these ideals is equal to a saturated ideal. In such a case to get the defining ideal of the multi-Rees algebra we only need to saturate the first syzygies of direct sum of this family of ideals. However, this fact is not true, when at least one of these ideals does not contain any nonzero divisor. In this paper we show that the defining ideal of the multi-Rees algebra of a family of ideals of a pol…

Local cohomology of multi-Rees algebras, Joint reduction numbers and product of complete ideals

We find conditions on the local cohomology modules of multi-Rees algebras of admissible filtrations which enable us to predict joint reduction numbers. As a consequence we are able to prove a generalisation of a result of Reid-Roberts-Vitulli in the setting of analytically unramified local rings for completeness of power products of complete ideals.

Rees Algebras and Almost Linearly Presented Ideals

Consider a grade 2 perfect ideal $I$ in $R=k[x_1,\cdots,x_d]$ which is generated by forms of the same degree. Assume that the presentation matrix $φ$ is almost linear, that is, all but the last column of $φ$ consist of entries which are linear. For such ideals, we find explicit forms of the defining ideal of the Rees algebra $\mathcal{R}(I)$. We also introduce the notion of iterated Jacobian duals.

Links of prime ideals and their Rees algebras

In a previous paper we exhibited the somewhat surprising property that most direct links of prime ideals in Gorenstein rings are equimultiple ideals with reduction number $1$. This led to the construction of large families of Cohen--Macaulay Rees algebras. The first goal of this paper is to extend this result to arbitrary Cohen--Macaulay rings. The means of the proof are changed since one cannot depend so heavily on linkage theory. We then study the structure of the Rees algebra of these links, more specifically we describe their canonical module in sufficient detail to be able to characterize…

The determinantal ideals of extended Hankel matrices

In this paper, we use the tools of Gröbner bases and combinatorial secant varieties to study the determinantal ideals $I_t$ of the extended Hankel matrices. Denote by $c$-chain a sequence $a_1,\...,a_k$ with $a_i+c<a_{i+1}$ for all $i=1,\...,k-1$. Using the results of $c$-chain, we solve the membership problem for the symbolic powers $I_t^{(s)}$ and we compute the primary decomposition of the product $I_{t_1}\... I_{t_k}$ of the determinantal ideals. Passing through the initial ideals and algebras we prove that the product $I_{t_1}\... I_{t_k}$ has a linear resolution and the multi-homogeneous…

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…

On ideals with the Rees property

A homogeneous ideal $I$ of a polynomial ring $S$ is said to have the Rees property if, for any homogeneous ideal $J \subset S $ which contains $I$, the number of generators of $J$ is smaller than or equal to that of $I$. A homogeneous ideal $I \subset S$ is said to be $\mathfrak m$-full if $\mathfrak mI:y=I$ for some $y \in \mathfrak m$, where $\mathfrak m$ is the graded maximal ideal of $S$. It was proved by one of the authors that $\mathfrak m$-full ideals have the Rees property and that the converse holds in a polynomial ring with two variables. In this note, we give examples of ideals whic…

行列式模的爆破代数

Blowup algebras of determinantal modules

本文研究由阶梯矩阵极大子式或单位区间行列式理想生成的模的爆破代数，确定其多重Rees代数及特殊纤维环的Gröbner基，并证明所得代数为Koszul Cohen–Macaulay正规整环，且在特征零时具有有理奇点、正特征时为F-有理。

On the Betti numbers and Rees algebras of ideals with linear powers

An ideal $I \subset \mathbb{k}[x_1, \ldots, x_n]$ is said to have linear powers if $I^k$ has a linear minimal free resolution, for all $k$. In this paper we study the Betti numbers of $I^k$, for ideals $I$ with linear powers. The Betti numbers are computed explicitly, as polynomials in $k$, for the ideal generated by all square free monomials of degree $d$, for $d=2, 3$ or $n-1$, and the product of all ideals generated by $s$ variables, for $s=n-1$ or $n-2$. We also study the generators of the Rees ideal, for ideals with linear powers. Especially, we are interested in ideals for which the Rees…

Cohen-Macaulayness of Rees Algebras of Diagonal Ideals

Given two determinantal rings over a field k. We consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the join variety. When the Rees algebra and the Symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.

Regularity, Rees algebra and Betti numbers of certain cover ideals

Let $S={\sf k}[X_1,\dots, X_n]$ be a polynomial ring, where ${\sf k}$ is a field. This article deals with the defining ideal of the Rees algebra of squarefree monomial ideal generated in degree $n-2$. As a consequence, we prove that Betti numbers of powers of the cover ideal of the complement graph of a tree do not depend on the choice of tree. Further, we study the regularity and Betti numbers of powers of cover ideals associated to certain graphs.

Cohen-Macaulayness of Rees Algebras of Modules

We provide the sufficient conditions for Rees algebras of modules to be Cohen-Macaulay, which has been proven in the case of Rees algebras of ideals by Johnson-Ulrich and Goto-Nakamura-Nishida. As it turns out the generalization from ideals to modules is not just a routine generalization, but requires a great deal of technical development. We use the technique of generic Bourbaki ideals introduced by Simis, Ulrich and Vasconcelos to obtain the Cohen-Macaulayness of Rees Algebras of modules.

关于理想的Rees代数与导子

On the Rees algebra and the conductor of an ideal

该研究引入Noetherian环中理想的导子作为Rees代数的研究工具，探究理想加元保持线性型的条件，推广Valla的结果，给出分次理想导子为极大理想时Rees与对称代数同维数的判据，并确定两类单项式理想的导子。

The strong Rees property of powers of the maximal ideal and Takahashi-Dao's question

In this paper, we introduce the notion of the strong Rees property (SRP) for $\mathfrak{m}$-primary ideals of a Noetherian local ring and prove that any power of the maximal ideal $\mathfrak{m}$ has its property if the associated graded ring $G$ of $\mathfrak{m}$ satisfies $\text{depth} \ G \ge 2$. As its application, we characterize two-dimensional excellent normal local domains so that $\mathfrak{m}$ is a $p_g$-ideal. Finally we ask what $\mathfrak{m}$-primary ideals have SRP and state a conjecture which characterizes the case when $\mathfrak{m}^n$ are the only ideals which have SRP.

三个变量中的Rees代数与几乎线性表示的理想

Rees algebra and almost linearly presented ideals in three variables

本文研究了三个变量中几乎线性表示的理想，在满足特定矩阵秩和$\Gs$条件下，推导出其Rees代数定义理想的显式公式。

Almost complete intersection binomial edge ideals and their Rees algebras

Let $G$ be a simple graph on $n$ vertices and $J_G$ denote the binomial edge ideal of $G$ in the polynomial ring $S = \mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the second graded Betti numbers of $J_G$, and we obtain a minimal presentation of it when $G$ is a tree or a unicyclic graph. We classify all graphs whose binomial edge ideals are almost complete intersection, prove that they are generated by a $d$-sequence and that the Rees algebra of their binomial edge ideal is Cohen-Macaulay. We also obtain an explicit description of the defining ideal of the Rees …

几类块图的二项边理想的符号幂与符号Rees代数

Symbolic Powers and Symbolic Rees Algebras of Binomial Edge Ideals of Some Classes of Block Graphs

本文研究几类块图的二项边理想的符号幂与符号Rees代数性质，证明垂悬团图的符号幂与普通幂一致，两类图的相关理想具符号$F$-分裂性，其符号Rees代数为强$F$-正则。

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。

Local Cohomology of Multi-Rees Algebras with Applications to Joint Reductions and Complete Ideals

In this paper, we obtain a generalization, in dimension $3$, of a theorem of David Rees about joint reductions of the bigraded filtration $\{ \overline{I^rJ^s}\}$ of complete ${\mathfrak m}$-primary ideals and vanishing of the second normal Hilbert coefficient $\overline{e}_2(IJ)$ where $R$ is a two-dimensional Cohen-Macaulay analytically unramified local ring with maximal ideal $\mathfrak m.$ This generalization is obtained as a consequence of a formula for the third local cohomology module of the extended Rees algebras of the $\mathbb Z^3$-graded filtration $\{\overline{I^rJ^sK^t}\}$ with su…

Rees algebras of square-free monomial ideals

We study the defining equations of the Rees algebra of square-free monomial ideals in a polynomial ring over a field. We determine that when an ideal $I$ is generated by $n$ square-free monomials of the same degree then $I$ has relation type at most $n-2$ as long as $n \leq 5$. In general, we establish the defining equations of the Rees algebra in this case. Furthermore, we give a class of examples with relation type at least $n-2$, where $n=μ(I)$. We also provide new classes of ideals of linear type. We propose the construction of a graph, namely the generator graph of an ideal, where the mon…

The almost Gorenstein Rees algebras of parameters

There is given a characterization for the Rees algebras of parameters in a Gorenstein local ring to be almost Gorenstein graded rings. A characterization is also given for the Rees algebras of socle ideals of parameters. The latter one shows almost Gorenstein Rees algebras rather rarely exist for socle ideals, if the dimension of the base local ring is greater than two.

Hilbert-Kunz function and Hilbert-Kunz multiplicity of some ideals of the Rees algebra

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…

The Rees Valuations of Complete Ideals in a Regular Local Ring

Let I be a complete m-primary ideal of a regular local ring (R,m). In the case where R has dimension two, the beautiful theory developed by Zariski implies that I factors uniquely as a product of powers of simple complete ideals and each of the simple complete factors of I has a unique Rees valuation. In the higher dimensional case, a simple complete ideal of R often has more than one Rees valuation, and a complete m-primary ideal I may have finitely many or infinitely many base points. For the ideals having finitely many base points, Lipman proves a unique factorization involving special star…

Almost Gorenstein Rees algebras of $p_g$-ideals, good ideals, and powers of the maximal ideals

Let $(A,{\mathfrak m})$ be a Cohen-Macaulay local ring and let $I$ be an ideal of $A$. We prove that the Rees algebra ${\mathcal R}(I)$ is an almost Gorenstein ring in the following cases: (1) $(A,{\mathfrak m})$ is a two-dimensional excellent Gorenstein normal domain over an algebraically closed field $K \cong A/{\mathfrak m}$ and $I$ is a $p_g$-ideal; (2) $(A,{\mathfrak m})$ is a two-dimensional almost Gorenstein local ring having minimal multiplicity and $I={\mathfrak m}^{\ell}$ for all $\ell \ge 1$; (3) $(A,{\mathfrak m})$ is a regular local ring of dimension $d \ge 2$ and $I={\mathfrak m}…

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。

Rees Algebras of Truncations of Complete Intersections

In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) onto it, where M is the direct sum of powers of m. We compute a Groebner basis of the ideal defining R(M). It turns out that the normal domain R(M) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well.

几乎完全交：幂的正则度与里斯代数

Almost Complete Intersections: Regularity of powers and Rees algebra

针对几乎完全交理想，给出满足特定条件的分次几乎完全交理想幂的正则度精确公式，研究相关里斯代数的极小双分次自由分解及对角线的Cohen-Macaulay性、Koszul性等性质。

On the almost Gorenstein property in Rees algebras of contracted ideals

The question of when the Rees algebra ${\mathcal R} (I)= \bigoplus_{n \ge 0}I^n$ of $I$ is an almost Gorenstein graded ring is explored, where $R$ is a two-dimensional regular local ring and $I$ a contracted ideal of $R$. It is known that ${\mathcal R} (I)$ is an almost Gorenstein graded ring for every integrally closed ideal $I$ of $R$. The main results of the present paper show that if $I$ is a contracted ideal with $\mathrm{o}(I) \le 2$, then ${\mathcal R} (I)$ is an almost Gorenstein graded ring, while if $\mathrm{o}(I) \ge 3$, then ${\mathcal R} (I)$ is not necessarily an almost Gorenstei…

Local cohomology modules of bigraded Rees algebras

Formulas are obtained in terms of complete reductions for the bigraded components of local cohomology modules of bigraded Rees algebras of 0-dimensional ideals in 2-dimensional Cohen-Macaulay local rings. As a consequence, cohomological expressions for the coefficients of the Bhattacharya polynomial of such ideals are obtained.

第一系数理想与Rees代数的$R_1$性质

First Coefficient ideals and $R_1$ property of Rees algebras

本文证明了在优正规局部环中，扩张Rees代数、Rees代数及其Proj的$R_1$性质等价，且等价于每个幂的理想整闭包等于其第一系数理想。

由二次型次极大生成的理想的 Rees 代数

Rees algebras of ideals submaximally generated by quadrics

本文研究由二次型次极大生成的 $\mathfrak{m}$-准素 Gorenstein 理想，证明其 Rees 代数为纤维型，并刻画特殊纤维环的定义理想。

When are the Rees algebras of parameter ideals almost Gorenstein graded rings?

Let $A$ be a Cohen-Macaulay local ring with $\operatorname{dim} A = d\ge 3$, possessing the canonical module ${\mathrm K}_A$. Let $a_1, a_2, \ldots, a_r$ $(3 \le r \le d)$ be a subsystem of parameters of $A$ and set $Q= (a_1, a_2, \ldots, a_r)$. It is shown that if the Rees algebra ${\mathcal R}(Q)$ of $Q$ is an almost Gorenstein graded ring, then $A$ is a regular local ring and $a_1, a_2, \ldots, a_r$ is a part of a regular system of parameters of $A$.

有限生成饱和多重Rees代数

Finitely generated saturated multi-Rees algebras

本文研究了饱和多重Rees代数的有限生成问题及相关长度函数的渐近行为，并在excellent local domains、二维local UFDs及多项式环商等设定下证明了有限生成性及长度函数的多项式或准多项式性质。

The depth of the Rees algebra of three general binary forms

One proves that the Rees algebra of an ideal generated by three general binary forms of same degree $\geq 5$ has depth one. The proof hinges on the behavior of the Ratliff-Rush filtration for low powers of the ideal and on establishing that certain large matrices whose entries are quadratic forms have maximal rank. One also conjectures a shorter result that implies the main theorem of the paper.

The $F$-pure threshold of a determinantal ideal

The $F$-pure threshold is a numerical invariant of prime characteristic singularities, that constitutes an analogue of the log canonical thresholds in characteristic zero. We compute the $F$-pure thresholds of determinantal ideals, i.e., of ideals generated by the minors of a generic matrix.

通过Rees代数研究的极良覆盖图

Very well-covered graphs via the Rees algebra

该研究针对Cohen-Macaulay极良覆盖图，利用与其覆盖理想关联的Rees代数，深入分析了这类图覆盖理想的代数性质，为相关代数图论研究提供了新的分析视角。

Rees algebras, Monomial Subrings and Linear Optimization Problems

In this thesis we are interested in studying algebraic properties of monomial algebras, that can be linked to combinatorial structures, such as graphs and clutters, and to optimization problems. A goal here is to establish bridges between commutative algebra, combinatorics and optimization. We study the normality and the Gorenstein property-as well as the canonical module and the a-invariant-of Rees algebras and subrings arising from linear optimization problems. In particular, we study algebraic properties of edge ideals and algebras associated to uniform clutters with the max-flow min-cut pr…

Initial algebras of determinantal rings, Cohen-Macaulay and Ulrich ideals

We study initial algebras of determinantal rings, defined by minors of generic matrices, with respect to their classical generic point. This approach leads to very short proofs for the structural properties of determinantal rings. Moreover, it allows us to classify their Cohen-Macaulay and Ulrich ideals.

Short polynomials in determinantal ideals

We show that a determinantal ideal generated by $t$-minors does not contain any nonzero polynomials with $t!/2$ or fewer terms. Geometrically this means that any nonzero polynomial vanishing on all matrices of rank at most $t-1$ has more than $t!/2$ terms.

Generating Residual Intersections of Determinantal Ideals

If $I$ is a perfect ideal in a local Cohen-Macaulay ring, the generators of ideals linked to $I$ are well understood. However, the generators of the residual intersections of $I$ have only been computed in a few special cases. In this paper, we show that the $n$-residual intersections of determinantal ideals of generic $2\times n$ matrices are sums of links.