Weak Lefschetz Property Research Articles

The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles

The Weak Lefschetz Property of Artinian Algebras Associated to Paths and Cycles

The weak Lefschetz property and mixed multiplicities of monomial ideals

The weak Lefschetz property and mixed multiplicities of monomial ideals

The weak Lefschetz property for Standard graded, Artinian Gorenstein algebras of embedding dimension four and socle degree three

Let k be an arbitrary field and A be a standard graded Artinian Gorenstein k-algebra of embedding dimension four and socle degree three. Then, except for exactly one exception, A has the weak Lefschetz property. Furthermore, the exception occurs only in characteristic two.

Forcing the weak Lefschetz property for equigenerated monomial ideals

We classify the minimal number of generators of artinian equigenerated monomial ideals I I such that k [ x 1 , … , x n ] / I \Bbbk [x_1,\ldots ,x_n]/I is forced to have the weak Lefschetz property.

Perazzo hypersurfaces and the weak Lefschetz property

We deal with Perazzo hypersurfaces X=V(f) in Pn+2 defined by a homogeneous polynomial f(x0,x1,…,xn,u,v)=p0(u,v)x0+p1(u,v)x1+⋯+pn(u,v)xn+g(u,v), where p0,p1,…,pn are algebraically dependent but linearly independent forms of degree d−1 in K[u,v] and g is a form in K[u,v] of degree d. Perazzo hypersurfaces have vanishing hessian and, hence, the associated graded artinian Gorenstein algebra Af fails the strong Lefschetz property. In this paper, we first determine the maximum and minimum Hilbert function of Af, we prove that the Hilbert function of Af is always unimodal and we determine when Af satisfies the weak Lefschetz property. We illustrate our results with many examples and we show that our results do not generalize to Perazzo hypersurfaces X=V(f) in Pn+3 defined by a homogeneous polynomial f(x0,x1,…,xn,u,v,w)=p0(u,v,w)x0+p1(u,v,w)x1+⋯+pn(u,v,w)xn+g(u,v,w), where p0,p1,…,pn are algebraically dependent but linearly independent forms of degree d−1 in K[u,v,w] and g is a form in K[u,v,w] of degree d.

Unique maximal Betti diagrams for Artinian Gorenstein k-algebras with the weak Lefschetz property

We give an alternate proof for a theorem of Migliore and Nagel. In particular, we show that if H is an SI-sequence, then the collection of Betti diagrams for all Artinian Gorenstein k-algebras with the weak Lefschetz property and Hilbert function H has a unique largest element.

On the Lefschetz property for quotients by monomial ideals containing squares of variables

Let Δ be an (abstract) simplicial complex on n vertices. One can define the Artinian monomial algebra A ( Δ ) = k [ x 1 , … , x n ] / 〈 x 1 2 , … , x n 2 , I Δ 〉 , where k is a field of characteristic 0 and I Δ is the Stanley-Reisner ideal associated to Δ . In this paper, we aim to characterize the Weak Lefschetz Property (WLP) of A ( Δ ) in terms of the simplicial complex Δ . We are able to completely analyze when the WLP holds in degree 1, complementing work by Migliore, Nagel and Schenck on the WLP for quotients by quadratic monomials. We give a complete characterization of all 2-dimensional pseudomanifolds Δ such that A ( Δ ) satisfies the WLP. We also construct Artinian Gorenstein algebras that fail the WLP by combining our results and the standard technique of Nagata idealization.

On the weak Lefschetz property for height four equigenerated complete intersections

We consider the conjecture that all artinian height 4 complete intersections of forms of the same degree d d have the Weak Lefschetz Property (WLP). We translate this problem to one of studying the general hyperplane section of a certain smooth curve in P 3 \mathbb P^3 , and our main tools are the Socle Lemma of Huneke and Ulrich together with a careful liaison argument. Our main results are (i) a proof that the property holds for d = 3 , 4 d=3,4 and 5; (ii) a partial result showing maximal rank in a non-trivial but incomplete range, cutting in half the previous unknown range; and (iii) a proof that maximal rank holds in a different range, even without assuming that all the generators have the same degree. We furthermore conjecture that if there were to exist any height 4 complete intersection generated by forms of the same degree and failing WLP then there must exist one (not necessarily the same one) failing by exactly one (in a sense that we make precise). Based on this conjecture we outline an approach to proving WLP for all equigenerated complete intersections in four variables. Finally, we apply our results to the Jacobian ideal of a smooth surface in P 3 \mathbb P^3 .

Free resolutions and Lefschetz properties of some Artin Gorenstein rings of codimension four

In (Stanley, 1978), Stanley constructs an example of an Artinian Gorenstein (AG) ring A with non-unimodal H-vector (1,13,12,13,1). Migliore-Zanello show in (Migliore and Zanello, 2017) that for regularity r=4, Stanley's example has the smallest possible codimension c for an AG ring with non-unimodal H-vector.The weak Lefschetz property (WLP) has been much studied for AG rings; it is easy to show that an AG ring with non-unimodal H-vector fails to have WLP. In codimension c=3 it is conjectured that all AG rings have WLP. For c=4, Gondim shows in (Gondim, 2017) that WLP always holds for r≤4 and gives a family where WLP fails for any r≥7, building on Ikeda's example (Ikeda, 1996) of failure for r=5. In this note we study the minimal free resolution of A and relation to Lefschetz properties (both weak and strong) and Jordan type for c=4 and r≤6.

Some notes and corrections of the paper “The non-Lefschetz locus”

A finite length graded R-module M has the Weak Lefschetz Property if there is a linear form ℓ in R such that the multiplication map ×ℓ:Mi→Mi+1 has maximal rank. The set of linear forms with this property form a Zariski-open set and its complement is called the non-Lefschetz locus. The main result is to give a complete proof for the theorem stated in [1] that for any general complete intersection I in R[x1,x2,x3,x4] the non-Lefschetz locus has expected codimension.

The non-Lefschetz locus of vector bundles of rank 2 over [formula omitted

A finite length graded R-module M has the Weak Lefschetz Property if there is a linear element ℓ in R such that the multiplication map ×ℓ:Mt→Mt+1 has maximal rank for every integer t. The set of linear forms with this property form a Zariski-open set and its complement is called the non-Lefschetz locus. In this paper we study the non-Lefschetz locus for the first cohomology module H⁎1(P2,E) of a vector bundle E of rank 2 over P2. The main result is to show that this non-Lefschetz locus has the expected codimension under the assumption that E is general.

A classification of the weak Lefschetz property for almost complete intersections generated by uniform powers of general linear forms

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, Mir\'o-Roig, and Nagel.

Perazzo 3-folds and the weak Lefschetz property

We deal with Perazzo 3-folds in P4, i.e. hypersurfaces X=V(f)⊂P4 of degree d defined by a homogeneous polynomial f(x0,x1,x2,u,v)=p0(u,v)x0+p1(u,v)x1+p2(u,v)x2+g(u,v), where p0,p1,p2 are algebraically dependent but linearly independent forms of degree d−1 in u,v, and g is a form in u,v of degree d. Perazzo 3-folds have vanishing hessian and, hence, the associated graded Artinian Gorenstein algebra Af fails the strong Lefschetz Property. In this paper, we determine the maximum and minimum Hilbert function of Af and we prove that if Af has maximal Hilbert function it fails the weak Lefschetz Property while it satisfies the weak Lefschetz Property when it has minimum Hilbert function. In addition, we classify all Perazzo 3-folds in P4 such that Af has minimum Hilbert function.

Lefschetz properties of some codimension three Artinian Gorenstein algebras

Codimension two Artinian algebras have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three Artinian Gorenstein algebras. Despite much work, the strong Lefschetz property for codimension three Artinian Gorenstein algebra has remained largely mysterious; our results build on and strengthen some of the previous results. We here show that every standard-graded codimension three Artinian Gorenstein algebra A having maximum value of the Hilbert function at most six has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is greater than the socle degree of A, we show that A is almost strong Lefschetz, they are strong Lefschetz except in the extremal pair of degrees.

Quadratically presented grade three Gorenstein ideals

Let J be a quadratically presented grade three Gorenstein ideal in the standard graded polynomial ring R=k[x,y,z], where k is a field. Assume that R/J satisfies the weak Lefschetz property. We give the presentation matrix for J in terms of the coefficients of a Macaulay inverse system for J. (This presentation matrix is an alternating matrix and J is generated by the maximal order Pfaffians of the presentation matrix.) Our formulas are computer friendly; they involve only matrix multiplication; they do not involve multilinear algebra or complicated summations. As an application, we give the presentation matrix for J1=(xn+1,yn+1,zn+1):(x+y+z)n+1, when n is even and the characteristic of k is zero. Generators for J1 had been identified previously; but the presentation matrix for J1 had not previously been known. The first step in our proof is to give improved formulas for the presentation matrix of a linearly presented grade three Gorenstein ideal I in terms of the coefficients of the Macaulay inverse system for I.

Lefschetz properties for jacobian rings of cubic fourfolds and other Artinian algebras

In this paper, we exploit some geometric-differential techniques to prove the strong Lefschetz property in degree 1 for a complete intersection standard Artinian Gorenstein algebra of codimension 6 presented by quadrics. We prove also some strong Lefschetz properties for the same kind of Artinian algebras in higher codimensions. Moreover, we analyze some loci that come naturally into the picture of “special” Artinian algebras: for them we give some geometric descriptions and show a connection between the non emptiness of the so-called non-Lefschetz locus in degree 1 and the “lifting” of a weak Lefschetz property to an algebra from one of its quotients.

The union of two linear star configurations in [formula omitted] all have generic Hilbert function

It has remained unknown if the union of two linear star configurations in P2 has generic Hilbert function. In this paper, we resolve this question, proving that the union of a linear star configuration X in P2 of type s and a fat point scheme Y in P2 of multiplicity (t−1) with s≥t≥2 has generic Hilbert function. As an application, we show that an Artinian ring R/(IX+IY) of two linear star configurations X and Y of codimension 2 in P3 of type s and t with s,t≥3 has the weak Lefschetz property.

Artinian algebras and Jordan type

There has been much work on strong and weak Lefschetz conditions for graded Artinian algebras A, especially those that are Artinian Gorenstein. A more general invariant of an Artinian algebra A or finite A-module M that we consider here is the set of Jordan types of elements of the maximal ideal 𝔪 of A, acting on M. Here, the Jordan type of ℓ∈𝔪A is the partition giving the Jordan blocks of the multiplication map mℓ:M→M. In particular, we consider the Jordan type of a generic linear element ℓ in A1, or in the case of a local ring 𝒜, that of a generic element ℓ∈𝔪𝒜, the maximum ideal. We often take M=A, the graded algebra, or M=𝒜 a local algebra. The strong Lefschetz property of an element, as well as the weak Lefschetz property can be expressed simply in terms of its Jordan type and the Hilbert function of M. However, there has not been until recently a systematic study of the set of possible Jordan types for a given Artinian algebra A or A-module M, except, importantly, in modular invariant theory, or in the study of commuting Jordan types. We first show some basic properties of the Jordan type. In a main result we show an inequality between the Jordan type of ℓ∈𝔪𝒜 and a certain local Hilbert function. In our last sections we give an overview of topics such as the Jordan types for Nagata idealizations, for modular tensor products, and for free extensions, including examples and some new results. We also propose open problems.

The weak Lefschetz property of graded Gorenstein algebras associated to the Apéry set of a numerical semigroup

It has been conjectured that all graded Artinian Gorenstein algebras of codimension three have the weak Lefschetz property over a field of characteristic zero. In this paper, we study the weak Lefschetz property of associated graded algebras A of the Apéry set of M-pure symmetric numerical semigroups generated by four natural numbers. These algebras are graded Artinian Gorenstein algebras of codimension three.

Togliatti systems associated to the dihedral group and the weak Lefschetz property

In this note, we study Togliatti systems generated by invariants of the dihedral group D2d acting on k[x0, x1, x2]. This leads to the first family of non-monomial Togliatti systems, which we call GT-systems with group D2d. We study their associated varieties \({S_{{D_{2d}}}}\), called GT-surfaces with group D2d. We prove that there are arithmetically Cohen-Macaulay surfaces whose homogeneous ideal, \(I({S_{{D_{2d}}}})\), is minimally generated by quadrics and we find a minimal free resolution of \(I({S_{{D_{2d}}}})\).