Mircea Cimpoeas
We compute the generic initial ideal of a complete intersection of embedding dimension three with strong Lefschetz property and we show that it is an almost reverse lexicographic ideal. This enable us to give a proof for Moreno's conjecture in the case $n=3$.
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We prove the strong Lefschetz property for certain complete intersections defined by products of linear forms, using a characterization of the strong Lefschetz property in terms of central simple modules.
In this paper, we study the strong Lefschetz property of artinian complete intersection ideals generated by products of linear forms. We prove the strong Lefschetz property for a class of such ideals with binomial generators.
In this paper we classify the monomial complete intersection algebras, in two variables, and of positive characteristic, which has the strong Lef- schetz property. Together with known results, this gives a complete classi- fication of the monomial complete intersections with the strong Lefschetz property.
Stanley proved that, in characteristic zero, all artinian monomial complete intersections have the strong Lefschetz property. We provide a positive characteristic complement to Stanley's result in the case of artinian monomial complete intersections generated by monomials all of the same degree, and also for arbitrary artinian monomial complete intersections in characteristic two. To establish these results, we first prove an a priori lower bound on the characteristics that guarantee the Lefschetz properties. We then use a variety of techniques to complete the classifications.
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.
A binary tree of complete intersections with the strong Lefschetz property
本文构造了由连续次数幂和对称多项式理想及其自然导出理想组成的二叉树形完全交族，并利用该结构证明所有成员均具有强 Lefschetz 性质。
The Non-Lefschetz Locus of Monomial Complete Intersections in Positive Characteristic
本文完整刻画了正特征域上不满足弱 Lefschetz 性质的单项式完全交集的指数向量轨迹，并用出租车度量下正象限格球的平移显式描述该轨迹。
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.
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。
For standard graded Artinian $K$-algebras defined by componentwise linear ideals and Gotzmann ideals, we give conditions for the weak Lefschetz property in terms of numerical invariants of the defining ideals.
In this paper we study the Lefschetz properties of monomial complete intersections in positive characteristic. We give a complete classification of the strong Lefschetz property when the number of variables is at least three, which proves a conjecture by Cook II. We also extend earlier results on the weak Lefschetz property by dropping the assumption on the residue field being infinite, and by giving new sufficient criteria.
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 the WLP and SLP of artinian monomial ideals in $S=\mathbb{K}[x_1,\dots ,x_n]$ via studying their minimal free resolutions. We study the Lefschetz properties of such ideals where the minimal free resolution of $S/I$ is linear for at least $n-2$ steps. We give an affirmative answer to a conjecture of Eisenbud, Huneke and Ulrich for artinian monomial ideals with almost linear resolutions.
We give a necessary and sufficient condition for a standard graded Artinian ring defined by an m-full ideal, to have the weak Lefschetz property in terms of graded Betti numbers. This is a generalization of a theorem of Wiebe for componentwise linear ideals. We also prove that the class of componentwise linear ideals and that of completely m-full ideals coincide in characteristic zero and in positive characteristic, with the assumption that the generic initial ideal w.r.t. the graded reverse lexicographic order is stable.
For a standard Artinian $k$-algebra $A=R/I$, we give equivalent conditions for $A$ to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of the generic initial ideal $\mathrm{gin}(I)$ of $I$ under the reverse lexicographic order. Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of $\mathrm{gin}(I)$ are determined just by the the Hilbert function of $I$ if $A$ has the weak Lefschetz property. Furthermore, for the case that $A$ is a standard Artinian $k$-algebra of codimens…
Forcing the weak Lefschetz property for equigenerated monomial ideals
本文研究 artinian 等生成单项式理想，刻画使对应商环必然具备弱 Lefschetz 性质所需的极小生成元个数。
We prove that any quadratic complete intersection with certain action of the symmetric group has the strong Lefschetz property over a field of characteristic zero. As a consequence of it we construct a new class of homogeneous complete intersections with generators of higher degrees which have the strong Lefschetz property.
Monomial ideals and the failure of the Strong Lefschetz property
该研究针对由d≥2次单项式生成的单项式理想的阿廷商环，给出了不满足强莱夫谢茨性质时d次希尔伯特函数的精确下界，还推导了其他理想类的下界并关联已有分类结果。
We determine a sharp lower bound for the Hilbert function in degree $d$ of a monomial algebra failing the weak Lefschetz property over a polynomial ring with $n$ variables and generated in degree $d$, for any $d\geq 2$ and $n\geq 3$. We consider artinian ideals in the polynomial ring with $n$ variables generated by homogeneous polynomials of degree $d$ invariant under an action of the cyclic group $\mathbb{Z}/d\mathbb{Z}$, for any $n\geq 3$ and any $d\geq 2$. We give a complete classification of such ideals in terms of the weak Lefschetz property depending on the action.
We introduce the $k$-strong Lefschetz property ($k$-SLP) and the $k$-weak Lefschetz property ($k$-WLP) for graded Artinian $K$-algebras, which are generalizations of the Lefschetz properties. The main results obtained in this paper are as follows: 1. Let $I$ be a graded ideal of $R=K[x_1, x_2, x_3]$ whose quotient ring $R/I$ has the SLP. Then the generic initial ideal of $I$ is the unique almost revlex ideal with the same Hilbert function as $R/I$. 2. Let $I$ be a graded ideal of $R=K[x_1, x_2, ..., x_n]$ whose quotient ring $R/I$ has the $n$-SLP. Suppose that all $k$-th differences of the Hil…
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.
We show that the Specht ideal of a two-rowed partition is perfect over an arbitrary field, provided that the characteristic is either zero or bounded below by the size of the second row of the partition, and we show this lower bound is tight. We also establish perfection and other properties of certain variants of Specht ideals, and find a surprising connection to the weak Lefschetz property. Our results in particular give a self-contained proof of Cohen-Macaulayness of certain $h$-equals sets, a result previously obtained by Etingof-Gorsky-Losev over the complex numbers using rational Cheredn…
The weak and strong Lefschetz properties are two basic properties that Artinian algebras may have. Both Lefschetz properties may vary under small perturbations or changes of the characteristic. We study these subtleties by proposing a systematic way of deforming a monomial ideal failing the weak Lefschetz property to an ideal with the same Hilbert function and the weak Lefschetz property. In particular, we lift a family of Artinian monomial ideals to finite level sets of points in projective space with the property that a general hyperplane section has the weak Lefschetz property in almost all…
On the weak and strong Lefschetz properties for initial ideals of determinantal ideals with respect to diagonal monomial orders
研究对角单项序下多项式环中t-子式生成理想的初始理想的Lefschetz性质，证明最大子式情形具有强Lefschetz性质，而t较小时在mn超过某界时弱Lefschetz性质不成立，并给出Murai问题的反例。
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 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 …
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。
We study the weak Lefschetz property of artinian Gorenstein algebras and in particular of artinian complete intersections. In codimension four and higher, it is an open problem whether all complete intersections have the weak Lefschetz property. For a given artinian Gorenstein algebra $A$ we ask what linear forms are Lefschetz elements for this particular algebra, i.e., which linear forms $\ell$ give maximal rank for all the multiplication maps $\times \ell: [A]_i \longrightarrow [A]_{i+1}$. This is a Zariski open set and its complement is the \emph{non-Lefschetz locus}. For monomial complete …
We define the strong Lefschetz property for finite graded modules over graded Artinian algebras whose grading is not necessarily standard. We show that most results which have been obtained for Artinian algebras with standard grading can be extended for non-standard grading. Our results on the strong Lefschetz property for non-standard grading can be used to prove that certain Artinian complete intersections with standard grading have the strong Lefschetz property.
We consider artinian algebras $A=\mathbb{C}[x_0,\ldots,x_m]/I$, with $I$ generated by a regular sequence of homogeneous forms of the same degree $d\geq 2$. We show that the multiplication by a general linear form from $A_{d-1}$ to $A_d$ is injective. We prove that the Weak Lefschetz Property holds for artinian complete intersection algebras as above, with $d=2$ and $m\leq 4$. Apparently, this was previously known only for $m\leq 3$. Although we are proposing only very limited progress towards the WLP conjecture for complete intersections, we hope that the methods of the present article can ill…
Let K be an algebraically closed field of characteristic p > 0. We apply a theorem of C. Han to give an explicit description for the weak Lefschetz property of the monomial Artinian complete intersection A = K[X,Y,Z]/(X^d,Y^d,Z^d) in terms of d and p. This answers a question of J. Migliore, R. M. Miro-Roig and U. Nagel and, equivalently, characterizes for which characteristics the rank-2 syzygy bundle Syz(X^d,Y^d,Z^d) on PP^2 satisfies the Grauert-Muelich theorem. As a corollary we obtain that for p=2 the algebra A has the weak Lefschetz property if and only if d=(2^t+1)/3 or d=(2^t-1)/3 for s…
The strong Lefschetz property for quadratic reverse lexicographic ideals
本文研究含平方变量与反字典序无平方二次项的理想，刻画其 Hilbert 级数，并证明其商代数在特征零及充分高正特征下具有强 Lefschetz 性质。
Given an ideal $I=(f_1,\ldots,f_r)$ in $\mathbb C[x_1,\ldots,x_n]$ generated by forms of degree $d$, and an integer $k>1$, how large can the ideal $I^k$ be, i.e., how small can the Hilbert function of $\mathbb C[x_1,\ldots,x_n]/I^k$ be? If $r\le n$ the smallest Hilbert function is achieved by any complete intersection, but for $r>n$, the question is in general very hard to answer. We study the problem for $r=n+1$, where the result is known for $k=1$. We also study a closely related problem, the Weak Lefschetz property, for $S/I^k$, where $I$ is the ideal generated by the $d$'th powers of the v…
On Strong Lefschetz Property of 0-dimensional Complete Intersections and Associated Forms
研究零维完全交集满足强Lefschetz性质的条件，证明其关联形式的Hessian非零，探讨关联形式的重建方法及其在Jacobian理想中的应用。
In [MMR], Michałek--Miró-Roig give a beautiful geometric characterization of Artinian quotients by ideals generated by quadratic or cubic monomials, such that the multiplication map by a general linear form fails to be injective in the first nontrivial degree. Their work was motivated by conjectures of Ilardi and Mezzetti-Miró-Roig-Ottaviani, connecting the failure to Laplace equations and classical results of Togliatti on osculating planes. We study quotients by quadratic monomial ideals, explaining failure of the Weak Lefschetz Property for some cases not covered by [MMR].
Stanley showed that monomial complete intersections have the strong Lefschetz property. Extending this result we show that a simple extension of an Artinian Gorenstein algebra with the strong Lefschetz property has again the strong Lefschetz property.
On the weak Lefschetz property for ideals generated by powers of general linear forms
该研究刻画了一般线性形式幂生成的几乎完全交的首项理想，给出平方、立方生成理想满足d次弱莱夫谢茨性质（WLP）的变量数n的界，并证明平方情形下的界是紧的。
In this paper, we study the dependence of the weak Lefschetz property of algebras defined by a special class of monomials ideals in a polynomial ring with coefficient in a field, to the characteristic of the base field.
The Lefschetz question asks if multiplication by a power of a general linear form, $L$, on a graded algebra has maximal rank (in every degree). We consider a quotient by an ideal that is generated by powers of linear forms. Then the Lefschetz question is, for example, related to the problem whether a set of fat points imposes the expected number of conditions on a linear system of hypersurfaces of fixed degree. Our starting point is a result that relates Lefschetz properties in different rings. It suggests to use induction on the number of variables, $n$. If $n = 3$, then it is known that mult…
We study the weak Lefschetz property and the Hilbert function of level Artinian monomial almost complete intersections in three variables. Several such families are shown to have the weak Lefschetz property if the characteristic of the base field is zero or greater than the maximal degree of any minimal generator of the ideal. Two of the families have an interesting relation to tilings of hexagons by lozenges. This lends further evidence to a conjecture by Migliore, Miro-Roig, and the second author. Finally, using our results about the weak Lefschetz property, we show that the Hilbert function…
In this paper we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Apéry set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.
We introduce a method for studying the Lefschetz properties for $k[x,y]$-modules based on the Lindström-Gessel-Viennot Lemma. In particular, we prove that certain modules over Artinian Clements-Lindström rings in characteristic zero have the strong Lefschetz property. In particular, we show that every homogeneous idea in a Clements-Lindström ring of embedding dimension two has the strong Lefschetz property. As an application, we study the strong Lefschetz property of type two monomial ideals of codimension three.
In this work a combinatorial approach towards the weak Lefschetz property is developed that relates this property to enumerations of signed perfect matchings as well as to enumerations of signed families of non-intersecting lattice paths in certain triangular regions. This connection is used to study Artinian quotients by monomial ideals of a three-dimensional polynomial ring. Extending a main result in the recent memoir [Boij, Migliore, Miró-Roig, Nagel, Zanello], we completely classify the quotients of type two that have the weak Lefschetz property in characteristic zero. We also derive resu…
We characterize the monomial complete intersections in three variables satisfying the Weak Lefschetz Property (WLP), as a function of the characteristic of the base field. Our result presents a surprising, and still combinatorially obscure, connection with the enumeration of plane partitions. It turns out that the rational primes p dividing the number, M(a,b,c), of plane partitions contained inside an arbitrary box of given sides a,b,c are precisely those for which a suitable monomial complete intersection (explicitly constructed as a bijective function of a,b,c) fails to have the WLP in chara…
A standard graded artinian monomial complete intersection algebra $A=\Bbbk[x_1,x_2,\ldots,x_n]/(x_1^{a_1},x_2^{a_2},\ldots,x_n^{a_n})$, with $\Bbbk$ a field of characteristic zero, has the strong Lefschetz property due to Stanley in 1980. In this paper, we give a new proof for this result by using only the basic properties of linear algebra. Furthermore, our proof is still true in the case where the characteristic of $\Bbbk$ is greater than the socle degree of $A$, namely $a_1+a_2+\cdots+a_n - n$.
We study the weak Lefschetz property of a class of graded Artinian Gorenstein algebras of codimension three associated in a natural way to the Apéry set of a numerical semigroup generated by four natural numbers. We show that these algebras have the weak Lefschetz property whenever the initial degree of their defining ideal is small.
Lefschetz properties of squarefree monomial ideals via Rees algebras
本文利用单项式理想 Rees 代数理论，将 Lefschetz 性质与无平方单项式理想及单纯复形联系起来，推广了 Postnikov 的混合欧拉数并证明其正性。
Using a connection to lozenge tilings of triangular regions, we establish an easily checkable criterion that guarantees the weak Lefschetz property of a quotient by a monomial ideal. It is also shown that each such ideal also has a semistable syzygy bundle.
Parseval-Rayleigh identities for homogeneous complete intersections
本文在任意正特征下证明齐次完全交留数映射的Parseval-Rayleigh恒等式，并借此概念性地证明了特征2域上一般齐次完全交具有强Lefschetz性质。
According to the Grothendieck-Lefschetz theorem from SGA 2, there are no nontrivial line bundles on the punctured spectrum $U_R$ of a local ring $R$ that is a complete intersection of dimension $\ge 4$. Dao conjectured a generalization for vector bundles $\mathscr{V}$ of arbitrary rank on $U_R$: such a $\mathscr{V}$ is free if and only if $\mathrm{depth}_R(\mathrm{End}_R(Γ(U_R, \mathscr{V}))) \ge 4$. We use deformation theoretic techniques to settle Dao's conjecture. We also present examples showing that its assumptions are sharp and draw consequences for splitting of vector bundles on complet…
We introduce a natural correspondence between artinian monomial almost complete intersections in three variables and punctured hexagonal regions. We use this correspondence to investigate the algebras for the presence of the weak Lefschetz property. In particular, we relate the field characteristics in which such an algebra fails to have the weak Lefschetz property to the prime divisors of the enumeration of signed lozenge tilings of the associated punctured hexagonal region. On the one side this allows us to establish the weak Lefschetz property in many new cases. For numerous classes of punc…
Lattice Paths, Lefschetz Properties, and Almkvist's Conjecture in Two Variables
本文研究非标准分次完全交 $A(m,n)$，证明当 $n=2$ 时其同时具有强 Lefschetz 与复 Hodge–Riemann 性质当且仅当 $m$ 为偶数，从而支持 Almkvist 单峰性猜想的强化形式。
The Weak Lefschetz Property for Tensor Products of Artinian Monomial Algebras and Its Applications to Lollipop Graphs
本文研究Artinian单项式代数张量积的弱Lefschetz性质，并据此分类了由棒棒糖图边理想定义的代数$A(L_{m,n})$的该性质。
Lefschetz properties for monomial complete intersections
本文通过上同调与表示论方法，给出了正特征域上单项式完全交弱Lefschetz性质的完整数值判据，并利用Renaud算法为强Lefschetz性质的已知分类提供了新证明。
On the Weak Lefschetz Property for certain ideals generated by powers of linear forms
研究由线性形式的幂生成的理想是否具有 Weak Lefschetz Property，针对网格点集族给出方网格一致幂情形的完整答案及非方网格的猜想，并描述 WLP 成立时的非 Lefschetz 轨迹。
We prove that the $d$-component of the generic initial ideal, with respect to the reverse lexicographic order, of an ideal generated by a regular sequence of homogeneous polynomials of degree $d$ is revlex in a particular, but important, case. Using this property, we compute the generic initial ideal for several complete intersection with strong Lefschetz property.
Let $A=\pmb k[x_1,...,x_n]/{(x_1^d,...,x_n^d)}$, where $\pmb k$ is an infinite field. If $\pmb k$ has characteristic zero, then Stanley proved that $A$ has the Weak Lefschetz Property (WLP). Henceforth, $\pmb k$ has positive characteristic $p$. If $n=3$, then Brenner and Kaid have identified all $d$, as a function of $p$, for which $A$ has the WLP. In the present paper, the analogous project is carried out for $4\le n$. If $4\le n$ and $p=2$, then $A$ has the WLP if and only if $d=1$. If $n=4$ and $p$ is odd, then we prove that $A$ has the WLP if and only if $d=kq+r$ for integers $k,q,d$ with …
We prove that a sequence $h$ of non-negative integers is the Hilbert function of some Artinian Gorenstein algebra with the strong Lefschetz property if and only if it is an SI-sequence. This generalizes the result by T. Harima which characterizes the Hilbert functions of Artinian Gorenstein algebras with the weak Lefschetz property. We also provide classes of Artinian Gorenstein algebras obtained from the ideal of points in $\mathbb{P}^n$ such that some of their higher Hessians have non-vanishing determinants. Consequently, we provide families of such algebras satisfying the SLP.
A characterization of binomial Macaulay dual generators for complete intersections
本文刻画了使相应 Artinian 代数成为完全交的二项式 Macaulay 对偶生成元，并据此证明特征 0 下此类完全交代数具有强 Lefschetz 性质。
On the Lefschetz Property for quotients by monomial ideals containing squares of variables
该研究通过单纯复形Δ刻画Artinian单项式代数A(Δ)的弱Lefschetz性质，补充了已有工作，给出2维伪流形的完整刻画，并构造出不满足该性质的Artinian Gorenstein代数。
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