Related papers: Weak Lefschetz property of equigenerated complete …
Our starting point is a basic problem in Hermite interpolation theory, namely determining the least degree of a homogeneous polynomial that vanishes to some specified order at every point of a given finite set. We solve this problem if the…
In this paper, we investigate the weak Lefschetz property for tensor products of Artinian monomial algebras and complete quadratic monomial algebras. As an application, we classify the weak Lefschetz property of the Artinian algebras…
Motivated by the foundational result that a monomial complete intersection has the strong Lefschetz property (SLP) in characteristic zero, it is natural to ask when monomial almost complete intersections have the SLP. In this paper, using…
An almost complete intersection ideal can be seen as a $d$-sequence ideal with the minimal number of generators being one more than its height. In this paper, we give exact formulas for the regularity of powers of graded almost complete…
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.…
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…
Consider ideals $I$ of the form \[ I=(x_1^2,\dots, x_n^2)+\mathrm{RLex}(x_ix_j) \] where $\mathrm{RLex}(x_ix_j)$ is the ideal generated by all the square-free monomials which are greater than or equal to $x_ix_j$ in the reverse…
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…
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…
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,…
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 Ap\'ery set of numerical semigroups. To this aim we also prove a general result about the…
We give a sharp lower bound for the Hilbert function in degree $d$ of artinian quotients $\Bbbk[x_1,\ldots,x_n]/I$ failing the Strong Lefschetz property, where $I$ is a monomial ideal generated in degree $d \geq 2$. We also provide sharp…
In the paper untitled "Laplace equations and the Weak Lefschetz Property" the authors highlight the link between rational varieties satisfying a Laplace equation and artinian ideals that fail the Weak Lefschetz property. Continuing their…
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…
A graph is called weakly perfect if its vertex chromatic number equals its clique number. Let $R$ be a ring and $I(R)^*$ be the set of all left proper non-trivial ideals of $R$. The intersection graph of ideals of $R$, denoted by $G(R)$, is…
Migliore-Mir\'o-Roig-Nagel [Trans. A.M.S. 2011, arXiv: 0811.1023] show that the weak Lefschetz property (WLP) can fail for an ideal I in K[x_1,x_2,x_3,x_4] generated by powers of linear forms. This is in contrast to the analogous situation…
We prove that in the polynomial ring $Q=\mathsf{k}[x,y,z,w]$, with $\mathsf{k}$ an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals $I$ such that $(x,y,z,w)^4\subseteq I \subseteq (x,y,z,w)^2$ can be…
In 1994, Orlik and Terao introduced a commutative Artinian analog S/I(A) of the Orlik-Solomon algebra of a hyperplane arrangement A to answer a question of Aomoto. A central topic of investigation in the study of Artinian algebras is the…
It is well known that a monomial complete intersection has the strong Lefschetz property in characteristic zero. This property is equivalent to the statement that any power of the sum of the variables is a maximal rank element on the…
The purpose of this note is to characterize the finite Hilbert functions which force all of their artinian algebras to enjoy the Weak Lefschetz Property (WLP). Curiously, they turn out to be exactly those (characterized by Wiebe in $[Wi]$)…