Related papers: Some algebras with the weak Lefschetz property

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…

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…

We show that an Artinian quotient of K[x, y, z] by an ideal I generated by powers of linear forms has the Weak Lefschetz property. If the syzygy bundle of I is semistable this follows from results of Brenner-Kaid; our proof works without…

Let K be an algebraically closed field of characteristic zero and let I=(f_1,...,f_n) be a homogeneous R_+-primary ideal in R:=K[X,Y,Z]. If the corresponding syzygy bundle Syz(f_1,...,f_n) on the projective plane is semistable, we show that…

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…

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.

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…

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…

MacMahon enumerated the plane partitions in an $a \times b \times c$ box. These are in bijection to lozenge tilings of a hexagon, to certain perfect matchings, and to families of non-intersecting lattice paths. In this work we consider more…

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…

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…

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…

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…

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.

We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy…

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…

In [MMR], Micha\l{}ek--Mir\'o-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…

Let $R=\mathbb K[x,y,z]$ be a standard graded polynomial ring where $\mathbb K$ is an algebraically closed field of characteristic zero. Let $M = \oplus_j M_j$ be a finite length graded $R$-module. We say that $M$ has the Weak Lefschetz…

Given a simple graph $G$, the artinian monomial algebra associated to $G$, denoted by $A(G)$, is defined by the edge ideal of $G$ and the squares of the variables. In this article, we classify some tadpole graphs $G$ for which $A(G)$ has or…

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…