On mixed Hessians and the Lefschetz properties

Abstract

We introduce a new type of Hessian matrix, that we call Mixed Hessian. The mixed Hessian is used to compute the rank of a multiplication map by a power of a linear form in a standard graded Artinian Gorenstein algebra. In particular we recover the main result of [MW] for identifying Strong Lefschetz elements, generalizing it also for Weak Lefschetz elements. This criterion is also used to give a new proof that Boolean algebras have the Strong Lefschetz Property (SLP). We also construct new examples of Artinian Gorenstein algebras presented by quadrics that does not satisfy the Weak Lefschetz Property (WLP); we construct minimal examples of such algebras and we give bounds, depending on the degree, for their existence. Artinian Gorenstein algebras presented by quadrics were conjectured to satisfy WLP in [MN1,MN2], and in a previous paper we construct the first counter-examples (see [GZ]). *Partially supported by the CAPES postdoctoral fellowship, Proc. BEX 2036/14-2.

The Jordan type of graded Artinian Gorenstein algebras

“…Higher order Hessians have been introduced in [MW] to control the SLP and used in [Go, GZ] to produce series of algebras failing SLP or WLP. More recently in [GZ2] mixed Hessians have been introduced to control both SLP and WLP, they are a generalization of higher order Hessians. We want highlight that there are important recent works in the study of Jordan types (see [IMM, AIK]).…”

Complete intersection Jordan types in height two

“…There have been many studies of graded Artinian Gorenstein algebras satisfying the strong or weak Lefschetz property (see [7] and the references cited there). Recently, there have been studies of more general questions about the Jordan type of pairs ( , A) (see [4,5,7,10,16] and references cited.) By a result of F.H.S.…”

“…We denote by R = k[x, y] the polynomial ring in two variables over k. We will consider Artinian Gorenstein (so by F.H.S. Macaulay's result complete intersection) algebras A = R/ Ann F , where F ∈ E = k[X, Y ] is the Macaulay dual generator of A. T. Maeno and J. Watanabe in 2009 introduced a method of using higher Hessians to determine the strong or weak Lefschetz properties of a graded Artinian algebra [16]; this was further developed and used by T. Maeno and Y. Numata [15] and by R. Gondim and colleagues [4,5,3]. The Hilbert function of a graded CI quotient A of R satisfies H(A) = T , a symmetric sequence of the form T = (1 0 , 2 1 , .…”

Jordan types with small parts for Artinian Gorenstein algebras of codimension three

“…Gorenstein algebras are Poincaré duality algebras [14] and thus natural algebraic objects to cohomology rings of smooth complex projective varieties. There has been many studies in the Lefschetz properties and Jordan types of Artinian Gorenstein algebras [2,4,5,7,15]. Gorenstein algebras of codimension two are complete intersections and they all satisfy the SLP.…”