Strong Lefschetz Property Research Articles

A binary tree of complete intersections with the strong Lefschetz property

In this paper we give a new family of complete intersections which have the strong Lefschetz property. The family consists of Artinian algebras defined by ideals generated by power sum symmetric polynomials of consecutive degrees and of certain ideals naturally derived from them. This family has a structure of a binary tree and this observation is a key to prove that all members in it have the strong Lefschetz property.

Perazzo n-folds and the weak Lefschetz property

In this paper, we determine the maximum hmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{max}$$\\end{document} and the minimum hmin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{min}$$\\end{document} of the Hilbert vectors of Perazzo algebras AF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_F$$\\end{document}, where F is a Perazzo polynomial of degree d in n+m+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+m+1$$\\end{document} variables. These algebras always fail the Strong Lefschetz Property. We determine the integers n, m, d such that hmax\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{max}$$\\end{document} (resp. hmin\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$h_{min}$$\\end{document}) is unimodal, and we prove that AF\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$A_F$$\\end{document} always fails the Weak Lefschetz Property if its Hilbert vector is maximum, while it satisfies the Weak Lefschetz Property if it is minimum, unimodal, and satisfies an additional mild condition. We determine the minimal free resolution of Perazzo algebras associated to Perazzo threefolds in P4\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}^4$$\\end{document} with minimum Hilbert vectors. Finally we pose some open problems in this context.

A theorem of Gordan and Noether via Gorenstein rings

Gordan and Noether proved in their fundamental theorem that an hypersurface X=V(F)subseteq {{mathbb {P}}}^n with nle 3 is a cone if and only if F has vanishing hessian (i.e. the determinant of the Hessian matrix). They also showed that the statement is false if nge 4, by giving some counterexamples. Since their proof, several others have been proposed in the literature. In this paper we give a new one by using a different perspective which involves the study of standard Artinian Gorenstein {{mathbb {K}}}-algebras and the Lefschetz properties. As a further application of our setting, we prove that a standard Artinian Gorenstein algebra R={{mathbb {K}}}[x_0,dots ,x_4]/J with J generated by a regular sequence of quadrics has the strong Lefschetz property. In particular, this holds for Jacobian rings associated to smooth cubic threefolds.

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.

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.

A new proof of Stanley’s theorem on the strong Lefschetz property

A standard graded artinian monomial complete intersection algebra $A=\Bbbk [x_1,\ldots ,x_n]/(x_1^{a_1},\ldots ,x_n^{a_n})$, with $\Bbbk $ a field of characteristic zero, has the strong Lefschetz property defined by Stanley in 1980. In this paper, we give

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.

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.

On decomposing monomial algebras with the Lefschetz properties

We introduce a general technique for decomposing monomial algebras which we use to study the Lefschetz properties. We apply our technique to various classes of algebras, including monomial almost complete intersections and Gorenstein algebras. In particular, we prove that Gorenstein codimension three algebras arising from numerical semigroups have the strong Lefschetz property. We also study the reverse of the splitting operation -- a gluing operation -- which gives a way to construct monomial algebras with the Lefschetz properties.

Infinitesimal Variation Functions for Families of Smooth Varieties

In this paper we introduce some variation functions associated to the rank of the Infinitesimal Variations of Hodge Structure for a family of smooth projective complex curves. We give some bounds and inequalities and, in particular, we prove that if X is a smooth plane curve, then, there exists a first order deformation xi in H^1(T_X) which deforms X as plane curve and such that xi cdot :H^0(omega _X)rightarrow H^1({mathcal O}_{X}) is an isomorphism. We also generalize the notions of variation functions to higher dimensional case and we analyze the link between IVHS and the weak and strong Lefschetz properties of the Jacobian ring of a smooth hypersurface.

Cohomological Blowups of Graded Artinian Gorenstein Algebras along Surjective Maps

Abstract We introduce the cohomological blowup of a graded Artinian Gorenstein algebra along a surjective map, which we term BUG (blowup Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blowup of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersections. We have included many examples throughout this manuscript.

Waring and cactus ranks and strong Lefschetz property for annihilators of symmetric forms

In this note we show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the Strong Lefschetz Property. This is the first example of an explicit dual form with these properties. For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition. Moreover, we determine the Waring rank, the cactus rank, the resolution and the Strong Lefschetz Property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables. We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

Connected sums of graded Artinian Gorenstein algebras and Lefschetz properties

In their paper [1], H. Ananthnarayan, L. Avramov, and W.F. Moore introduced a connected sum construction for local Gorenstein rings A,B over a local Gorenstein ring T, which, in the graded Artinian case, can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative descriptions of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum of A,B over an Artinian Gorenstein algebra T preserves the weak or strong Lefschetz property, thus providing new classes of rings which satisfy these properties.

The Lefschetz question for ideals generated by powers of linear forms in few variables

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 multiplication by L always has maximal rank. We show that the same is true for multiplication by L2 if all linear forms are general. Furthermore, we give a complete description of when multiplication by L3 has maximal rank (and its failure when it does not). As a consequence, for such ideals that contain a quadratic or cubic generator, we establish results on the so-called strong Lefschetz property for ideals in n=3 variables, and the weak Lefschetz property for ideals in n=4 variables.

The resolution of (xN,yN,zN,wN)

Let k be a field of characteristic zero, n<N be positive integers, P be the polynomial ring k[x,y,z,w], F be the homogeneous polynomial xn+yn+zn+wn, K be the ideal (xN,yN,zN,wN), and P‾ be the hypersurface ring P‾=P/(F). We describe the minimal multi-homogeneous resolution of P‾/KP‾ by free P‾-modules, the socle degrees of P‾/KP‾, and the minimal multi-homogeneous resolution of the Gorenstein ring P/(K:F) by free P-modules. Our arguments use Stanley's theorem that every Artinian monomial complete intersection over a polynomial ring with coefficients from a field of characteristic zero has the strong Lefschetz property as well as a multi-grading on P for which both ideals K and (F) are homogeneous. The resolution of P‾/KP‾ by free P‾-modules is obtained from a Differential Graded Algebra resolution of P/(K:F) by free P-modules, together with one homotopy map. The multi-grading is used to prove that the resulting resolution is minimal.

Lefschetz Properties in Algebra, Geometry and Combinatorics

The themes of the workshop are the Weak Lefschetz Property – WLP – and the Strong Lefschetz Property – SLP. The name of these properties, referring to Artinian algebras, is motivated by the Lefschetz theory for projective manifolds, initiated by S. Lefschetz, and well established by the late 1950’s. In fact, Lefschetz properties of Artinian algebras are algebraic generalizations of the Hard Lefschetz property of the cohomology ring of a smooth projective complex variety. The investigation of the Lefschetz properties of Artinian algebras was started in the mid 1980’s and nowadays is a very active area of research. Although there were limited developments on this topic in the 20th century, in the last years this topic has attracted increasing attention from mathematicians of different areas, such as commutative algebra, algebraic geometry, combinatorics, algebraic topology and representation theory. One of the main features of the WLP and the SLP is their ubiquity and the quite surprising and still not completely understood relations with other themes, including linear configurations, interpolation problems, vector bundle theory, plane partitions, splines, d -webs, differential geometry, coding theory, digital image processing, physics and the theory of statistical designs, etc. among others.

The eigenvalues of the Hessian matrices of the generating functions for trees with k components

Let us consider a truncated matroid MΓr of rank r of a graphic matroid of a graph Γ. The basis for MΓr is the set of the forests with r edges in Γ. We consider this basis generating function and compute its Hessian. In this paper, we show that the Hessian of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph does not vanish by calculating the eigenvalues of the Hessian matrix. Moreover, we show that the Hessian matrix of the basis generating function of the truncated matroid of the graphic matroid of the complete or complete bipartite graph has exactly one positive eigenvalue. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the truncated matroid.

Monomial ideals and the failure of the Strong Lefschetz property

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 ge 2. We also provide sharp lower bounds for other classes of ideals, and connect our result to the classification of the Hilbert functions forcing the Strong Lefschetz property by Zanello and Zylinski.

Hilbert functions of Artinian Gorenstein algebras with the strong Lefschetz property

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 generali ...

The eigenvalues of Hessian matrices of the complete and complete bipartite graphs

In this paper, we consider the Hessian matrices $$H_{\Gamma }$$ of the complete and complete bipartite graphs, and the special value of $${\tilde{H}}_{\Gamma }$$ at $$x_{i}=1$$ for all $$x_{i}$$ . We compute the eigenvalues of $${\tilde{H}}_{\Gamma }$$ . We show that one of them is positive and that the others are negative. In other words, the metric with respect to the symmetric matrix $$\tilde{H}_{\Gamma }$$ is Lorentzian. Hence those Hessian $$\det (H_{\Gamma })$$ are not identically zero. As an application, we show the strong Lefschetz property for the Artinian Gorenstein algebra associated to the graphic matroids of the complete and complete bipartite graphs with at most five vertices.