Bahman Engheta
Let $R$ be a polynomial ring over a field of characteristic zero and let $I \subset R$ be a graded ideal of height $N$ which is minimally generated by $N+1$ homogeneous polynomials. If $I=(f_1,...,f_{N+1})$ where $f_i$ has degree $d_i$ and $(f_1,...,f_N)$ has height $N$, then the multiplicity of $R/I$ is bounded above by $\prod_{i=1}^N d_i - \max\{1, \sum_{i=1}^N (d_i-1) - (d_{N+1}-1) \}$.
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We establish the upper bound in the multiplicity conjecture of Herzog, Huneke and Srinivasan for the codimension three almost complete intersections. We also give some partial results in the case where I is the aci linked to a complete intersection in one step.
Given a finitely generated module $M$ over a local ring $A$ of characteristic $p$ with $\pd M < \infty$, we study the asymptotic intersection multiplicity $χ_\infty(M, A/\underline{x})$, where $\underline{x} = (x_1, \ldots, x_r)$ is a system of parameters for $M$. We show that there exists a system of parameters such that $χ_\infty$ is positive if and only if $\dim \Ext^{d-r}(M, A) = r$, where $d = \dim A$ and $r = \dim M$. We use this to prove several results relating to the Grade Conjecture, which states that $\grade M + \dim M = \dim A$ for any module $M$ with $\pd M < \infty$.
In this paper, we investigate the multiplicities and the log canonical thresholds of abelian quotient complete intersection singularities in term of the special datum. Moreover we give bounds of the multiplicity of abelian quotient complete intersection singularities.
We show that the Hilbert-Kunz multiplicity of a $d$-dimensional nonregular complete intersection over the algebraic closure of $F_p$, $p>2$ prime, is bounded by below by the Hilbert-Kunz multiplicity of the hypersurface $\sum _{i=0}^{d} x_i^2=0$, answering positively a conjecture of Watanabe and Yoshida in the case of complete intersections.
We show that Serre's Intersection Multiplicity Conjecture holds for a formal power series ring A over a complete, two-dimensional regular local ring R. From this, we deduce the corresponding result for the local rings of any scheme X which is a smooth extension of a regular, two-dimensional base Y. We also investigate the connection between intersection multiplicity and transversality in the unramified setting via a local analysis on the blowup.
We describe here some recent progress pertaining to the Serre Intersection Multiplicity Conjecture. In particular, we show that if A is an unramified regular local ring, then just as in the equicharacteristic case, the intersection multiplicity of two complimentary-dimensional modules is bounded below by the product of their Hilbert-Samuel multiplicities. We also explain, in terms of the blowup of Spec(A) at the closed point, the geometric significance of achieving this lower bound.
Let $R$ be a Noetherian local ring. We define the minimal $j$-multiplicity and almost minimal $j$-multiplicity of an arbitrary $R$-ideal on any finite $R$-module. For any ideal $I$ with minimal $j$-multiplicity or almost minimal $j$-multiplicity on a Cohen-Macaulay module $M$, we prove that under some residual assumptions, the associated graded module ${\rm gr}_I(M)$ is Cohen-Macaulay or almost Cohen-Macaulay, respectively. Our work generalizes the results for minimal multiplicity and almost minimal multiplicity obtained by Sally, Rossi, Valla, Wang, Huckaba, Elias, Corso, Polini, and VazPinto…
The Tjurina number of a plane curve with two branches and high intersection multiplicity
本文构造了一族具有两个分支的约化平面曲线，证明其Tjurina数在等奇异性类中恒定，并给出基于拓扑数据的封闭公式。
The aim of this note is to describe how to compute the intersection multiplicity defined by Jean Pierre Serre.
A Complete Characterization of Realizable (Embedding Dimension, Multiplicity) Pairs for Complete Intersection Numerical Semigroups
该研究解决了数值半群关系理论中的开放问题，给出了完全交数值半群的可实现（嵌入维数，重数）对的完整刻画，明确了对应的正整数对满足的条件。
In this short note, we confirm a conjecture of Vasconcelos which states that the Rees algebra of any Artinian almost complete intersection monomial ideal is almost Cohen-Macaulay.
We construct a complex of free-modules over a Gorenstein ring R of dimension five, for which the Euler characteristic and Dutta multiplicity are different. This complex is the resolution of an R-module of finite length and finite projective dimension. As a consequence, the ring R has a nonzero Todd class tau_3(R) and a bounded free complex whose local Chern character does not vanish on this class. In the course of our work, we construct a module N of finite length and finite projective dimension over the hypersurface A=K[u,v,w,x,y,z]/(ux+vy+wz), such that the Serre intersection multiplicity of…
Let $I$ be a monomial almost complete intersection ideal of a polynomial algebra $S$ over a field. Then Stanley's Conjecture holds for $S/I$ and $I$.
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 $3$.
We are interested in the structure of almost complete intersection ideals of grade 3. We give three constructions of these ideals and their free resolutions: one from the commutative algebra point of view, an equivariant construction giving a nice canonical form, andfinally an interpretation in terms of open sets in certain Schubert varieties.
Let $K$ be a field and $I$ a monomial ideal of the polynomial ring $S=K[x_1,\ldots, x_n]$. We show that if either: 1) $I$ is almost complete intersection, 2) $I$ can be generated by less than four monomials; or 3) $I$ is the Stanley-Reisner ideal of a locally complete intersection simplicial complex on $[n]$, then Stanley's conjecture holds for $S/I$.
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。
In this paper we examine the role of four Hilbert functions in the determination of the defining relations of the Rees algebra of almost complete intersections of finite colength. Because three of the corresponding modules are Artinian, some of these relationships are very effective, opening up tracks to the determination of the equations and also to processes of going from homologically defined sets of equations to higher degrees ones assembled by resultants.
Let $S$ be a polynomial ring over an algebraic closed field $k$ and $ \mathfrak p =(x,y,z,w) $ a homogeneous height four prime ideal. We give a finite characterization of the degree two component of ideals primary to $\mathfrak p$, with multiplicity $e \leq 3$. We use this result to give a tight bound on the projective dimension of almost complete intersections generated by five quadrics with $e \leq 3$.
We use Macaulay's inverse system to study the Hilbert series for almost complete intersections generated by uniform powers of general linear forms. This allows us to give a classification of the Weak Lefschetz property for these algebras, settling a conjecture by Migliore, Miró-Roig, and Nagel.
In this article, we propose a question on the annihilators of positive Koszul homologies of a system of parameters of an almost complete intersection $R$. The question can be stated in terms of the acyclicity of certain (finite) residual approximation complexes whose $0$-th homologies are the residue field of $R$. We show that our question has an affirmative answer for the first Koszul homology of any almost complete intersection, as well as for all positive Koszul homologies of certain system of parameters which exist in some almost complete intersection rings with small multiplicities. The s…
Let $R = S/I$ be a quotient of a standard graded polynomial ring $S$ by an ideal $I$ generated by quadrics. If $R$ is Koszul, a question of Avramov, Conca, and Iyengar asks whether the Betti numbers of $R$ over $S$ can be bounded above by binomial coefficients on the minimal number of generators of $I$. Motivated by previous results for Koszul algebras defined by three quadrics, we give a complete classification of the structure of Koszul almost complete intersections and, in the process, give an affirmative answer to the above question for all such rings.
Monomial Almost Complete Intersections and the Strong Lefschetz Property
本文以 Hilbert 级数为工具研究单项式几乎完全交的 SLP，完整分类了非纯幂生成元支集于两个变量的情形，并证明对称 Hilbert 级数情形总具有 SLP。
The subject matter is the structure of the Rees algebra of almost complete intersection ideals of finite colength in low-dimensional polynomial rings over fields. The main tool is a mix of Sylvester forms and iterative mapping cone construction. The material developed spins around ideals of forms in two or three variables in the search of those classes for which the corresponding Rees ideal is generated by Sylvester forms and is almost Cohen--Macaulay. A main offshoot is in the case where the forms are monomials. Another consequence is a proof that the Rees ideals of the base ideals of certain…
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.
We describe prime ideals of height 2 minimally generated by 3 elements in a Gorenstein, Nagata local ring of Krull dimension 3 and multiplicity at most 3. This subject is related to a conjecture of Y. Shimoda and to a long-standing problem of J. Sally.
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.
Using invariants from commutative algebra to count geometric objects is a basic idea in singularities. For example, the multiplicity of an ideal is used to count points of intersection of two analytic sets at points of non-transverse intersection. A problem with the multiplicity of an ideal or module is that it is only defined for modules and ideals of finite colength. In this paper we use pairs of modules and their multiplicities as a way around this difficulty. A key tool is the multiplicity-polar theorem which applies to families of pairs of modules, linking the multiplicity at a general pa…
Set-Theoretically Perfect Ideals and Residual Intersections
本文研究满足Serre条件Sₛ环中的代数剩余交，证明其存在自由逼近（即具相同根的完美子理想），由此确定剩余交重数的一致上界；在正特征下，剩余交是上同调完全交，故其簇在余维1连通。
Betti numbers and almost complete intersection monomial ideals
本文给出几乎完全交单项式理想的 Betti 数显式公式，用于快速构造极小自由分解，并刻画其 Cohen-Macaulay 性，同时推广至主单项式理想。
In this paper, we investigate the behavior of almost reverse lexicographic ideals with the Hilbert function of a complete intersection. More precisely, over a field $K$, we give a new constructive proof of the existence of the almost revlex ideal $J\subset K[x_1,\dots,x_n]$, with the same Hilbert function as a complete intersection defined by $n$ forms of degrees $d_1\leq \dots \leq d_n$. Properties of the reduction numbers for an almost revlex ideal have an important role in our inductive and constructive proof, which is different from the more general construction given by Pardue in 2010. We…
Bounds on multiplicity of MCM modules having non-extremal growth of betti-numbers
针对 Cohen-Macaulay 局部环上的极大 Cohen-Macaulay 模，在环为非完全交和完全交两种情形下，分别给出了满足特定 Betti 数曲率条件的模的重数下界，并通过例子验证结果的最优性。
Let $Λ$ be a numerical semigroup, $\mathcal{C}\subseteq \mathbb{A}^n$ the monomial curve singularity associated to $Λ$, and $\mathcal{T}$ its tangent cone. In this paper we provide a sharp upper bound for the least positive integer in $Λ$ in terms of the codimension and the maximum degree of the equations of $\mathcal{T}$, when $\mathcal{T}$ is not a complete intersection. A special case of this result settles a question of J. Herzog and D. Stamate.
In this note we first give a new bound on $e_{HK}(\sim)$ the Hilbert-Kunz multiplicity of invariant rings, by the help of the Noether's bound. Then, we simplify, extend and present applications of the reciprocity formulae due to L. Smith. He proved the formula over polynomial rings and his result is tight in the following sense: Over complete intersection rings with isolated singularity we show that the reciprocity formulae "$e_{HK}(R/I)+ e_{HK}(R/J) = e_{HK}(R/\underline{f})$" is equivalent with $\pd(I)<\infty$ when $I$ is an $\fm$-primary unmixed ideal linked to $J$ along with a regular sequ…
Let $R$ be a polynomial ring over a field. We prove an upper bound for the multiplicity of $R/I$ when $I$ is a homogeneous ideal of the form $I=J+(F)$, where $J$ is a Cohen-Macaulay ideal and $F\notin J$. The bound is given in terms of two invariants of $R/J$ and the degree of $F$. We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.
Conditions of multiplicity and applications for almost Gorenstein graded rings
本文证明了在重数和Cohen-Macaulay型条件下，几乎Gorenstein性质可推出Gorenstein性质，并应用于张量积及仿射半群环类。
Deciding the presence of the weak Lefschetz property often is a challenging problem. In this work an in-depth study is carried out in the case of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.
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 …
CI-sequences and almost complete intersections
本文研究几乎完全交阿廷代数的希尔伯特函数与分次贝蒂数，证明完全交的希尔伯特函数均可由几乎完全交实现，并在余维数3情形下分类了可能的贝蒂数与极小自由消解中的消去。
Let $R$ be a standard graded Noetherian algebra over an Artinian local ring. Motivated by the work of Achilles and Manaresi in intersection theory, we first express the multiplicity of $R$ by means of local $j$-multiplicities of various hyperplane sections. When applied to a homogeneous inclusion $A\subseteq B$ of standard graded Noetherian algebras over an Artinian local ring, this formula yields the multiplicity of $A$ in terms of that of $B$ and of local $j$-multiplicities of hyperplane sections along ${\rm Proj}\,(B)$. Our formulas can be used to find the multiplicity of special fiber ring…
We provide a structure theorem for all almost complete intersection ideals of depth three in any Noetherian local ring. In particular, we find that the minimal generators are the pfaffians of suitable submatrices of an alternating matrix. From the graded version of the previous result, we characterize the graded Betti numbers of all 3-codimensional almost complete intersection schemes of P^r.
We continue the study of intersection algebras $\mathcal B = \mathcal B_R(I, J)$ of two ideals $I, J$ in a commutative Noetherian ring $R$. In particular, we exploit the semigroup ring and toric structures in order to calculate various invariants of the intersection algebra when $R$ is a polynomial ring over a field and $I,J$ are principal monomial ideals. Specifically, we calculate the $F$-signature, divisor class group, and Hilbert-Samuel and Hilbert-Kunz multiplicities, sometimes restricting to certain cases in order to obtain explicit formulæ. This provides a new class of rings where formu…
Let $R$ be a local complete intersection and $M,N$ are $R$-modules such that $\ell(\Tor_i^R(M,N))<\infty$ for $i\gg 0$. Imitating an approach by Avramov and Buchweitz, we investigate the asymptotic behavior of $\ell(\Tor_i^R(M,N))$ using Eisenbud operators and show that they have well-behaved growth. We define and study a function $η^R(M,N)$ which generalizes Serre's intersection multiplicity $χ^R(M,N)$ over regular local rings and Hochster's function $θ^R(M,N)$ over local hypersurfaces. We use good properties of $η^R(M,N)$ to obtain various results on complexities of $\Tor$ and $\Ext$, vanish…
The second Veronese ideal $I_n$ contains a natural complete intersection $J_n$ generated by the principal $2$-minors of a symmetric $(n\times n)$-matrix. We determine subintersections of the primary decomposition of $J_n$ where one intersectand is omitted. If $I_n$ is omitted, the result is the other end of a complete intersection link as in liaison theory. These subintersections also yield interesting insights into binomial ideals and multigraded algebra. For example, if $n$ is even, $I_n$ is a Gorenstein ideal and the intersection of the remaining primary components of $J_n$ equals $J_n+\lan…
On the Regularity of Dominant and Almost Complete Intersection Monomial Ideals
本文针对几乎完全交单项式理想给出了Castelnuovo-Mumford正则性的显式公式，并证明了主导理想和几乎完全交理想满足Küronya-Pintye猜想。
We present a class of toric varieties $V$ which, over any algebraically closed field of characteristic zero, are defined by codim $V$+1 binomial equations.
We describe a class of affine toric varieties $V$ that are set-theoretically minimally defined by codim $V+1$ binomial equations over fields of any characteristic.
In 2012, Migliore, the first author, and Nagel conjectured that, for all $n\geq 4$, the artinian ideal $I=(L_0^d,\ldots,L_{2n+1}^d) \subset R=k[x_0,\ldots,x_{2n}]$ generated by the $d$-th powers of $2n+2$ general linear forms fails to have the weak Lefschetz property if and only if $d>1$. This paper is entirely devoted to prove partially this conjecture. More precisely, we prove that $R/I$ fails to have the weak Lefschetz property, provided $4\leq n\leq 8,\ d\geq 4$ or $d=2r,\ 1\leq r\leq 8,\ 4\leq n\leq 2r(r+2)-1$.
We study numerical semigroups with the property "multiplicity= embedding dimension+1", generated by concatenation of arithmetic sequences.
In this article we give explicit descriptions of the multiplicities of some classes of monomial ideals. For instance, we give a formula for the multiplicities of all codimension 1 monomial ideals, and another formula for the multiplicities of almost complete intersections. We also introduce a new class of ideals that extends the family of monomial complete intersections, and give a formula for their multiplicity, as well as a visual interpretation of this invariant.
An upper bound for the multiplicity and Wilf's conjecture for one-dimensional Cohen-Macaulay rings
本文研究一维 Cohen-Macaulay 环的重数上界，刻画等号情形，联系数值半群环的 Wilf 猜想，并证明近似 Gorenstein 环上的相应猜想。
(Almost) Complete Intersection Lovász-Saks-Schrijver ideals and regularity of their powers
本文研究了特定图类的 LSS-理想完全交集性质，并引入了 twisted 正则匹配分解数来判断其完全交集条件。
The generators of a colon ideal with an application to the weak Lefschetz property for monomial almost complete intersections in three variables
本文通过研究列理想生成元，探讨了三项式近完全交集的弱Lefschetz性质，并验证了一个猜想在特定情况下的正确性。
In this paper we investigate the question of when the determinantal ring $R$ over a field $k$ is an almost Gorenstein local/graded ring in the sense of Goto, Takahashi, and the author. As a consequence of the main result, we see that if $R$ is a non-Gorenstein almost Gorenstein local/graded ring, then the ring $R$ has a minimal multiplicity.
Let $(R,\mathfrak m)$ be a local ring of characteristic $p>0$ and $M$ a finitely generated $R$-module. In this note we consider the limit: $\lim_{n\to \infty} \frac{\ell(H^0_{\mathfrak m}(F^n(M)))}{p^{n\dim R}} $ where $F(-)$ is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when $R$ is excellent and has isolated singularity. Furthermore, if $R$ is a complete intersection, then the limit is 0 if and only if the projective dimension of $M$ is less than the Krull dimension of $R$. We exploit this fact to give a quick proof that if $R$ is a comple…
Many algebras are expected to have the Weak Lefschetz property though this is often very difficult to establish. We illustrate the subtlety of the problem by studying monomial and some closely related ideals. Our results exemplify the intriguing dependence of the property on the characteristic of the ground field, and on arithmetic properties of the exponent vectors of the monomials.
A study of a quadratic almost complete intersection ideal and its linked Gorenstein ideal
本文研究了二次几乎完全交理想及其链接的Gorenstein理想，计算了其Betti数，并揭示了相关环的socle结构和维度特性。
The structure of almost symmetric almost complete intersection numerical semigroups
本文证明几乎对称且几乎完全交的数值半群具有级联矩阵结构，并由此得到刚性定理、级联多项式及Herzog–Watanabe问题的肯定解答。
A current research theme is to compare symbolic powers of an ideal $I$ with the regular powers of $I$. In this paper, we focus on the case that $I=I_X$ is an ideal defining an almost complete intersection (ACI) sets of points $X$ in $\mathbb{P}^1\times\mathbb{P}^1$. In particular, we describe a minimal free bigraded resolution of a non arithmetically Cohen-Macaulay (also non homogeneus) set of fat points $\mathcal Z$ whose support is an ACI. We call $\mathcal Z$ a fat ACI. We also show that its symbolic and ordinary powers are equal, i.e, $I_{\mathcal Z}^{(m)}=I_{\mathcal Z}^{m}$ for any $m\ge…
We generalize some properties related to Hilbert series and Lefschetz properties of Milnor algebras of projective hypersurfaces with isolated singularities to the more general case of an almost complete intersection ideal $J$ of dimension one. When the saturation $I$ of $J$ is a complete intersection, we get explicit formulas for a number of related invariants. New examples of hypersurfaces $V:f=0$ in $P^n$ whose Jacobian ideal $J_f$ satisfies this property and with explicit nontrivial Alexander polynomials are given in any dimension. A Lefschetz type property for the graded quotient $I/J$ is …
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