Weak Lefschetz property for codimension-three Artinian Gorenstein algebras

Let $\Bbbk$ be a field of characteristic zero and let

$$A=\Bbbk[x,y,z]/I$$

be a standard graded Artinian Gorenstein algebra of embedding dimension three. The algebra $A$ has the weak Lefschetz property if there is a linear form $\ell\in A_1$ such that every multiplication map

$$\times\ell:A_i\longrightarrow A_{i+1}$$

has maximal rank.

Conjecture. Every standard graded Artinian Gorenstein algebra of embedding dimension three over a field of characteristic zero has the weak Lefschetz property.

By Macaulay duality, such an algebra can be written as

$$A\cong\Bbbk[x,y,z]/\operatorname{Ann}(F),$$

where $F\in\Bbbk[X,Y,Z]$ is a homogeneous ternary form and $x,y,z$ act by differentiation. Thus a prospective counterexample is specified by finitely many coefficients. If

$$\ell=ax+by+cz,$$

then the matrices of the maps $\times\ell$ have entries polynomial—in fact linear—in $a,b,c$. The weak Lefschetz property asks whether some $[a:b:c]\in\mathbb P^2$ makes all these matrices have maximal rank. Failure can therefore be expressed exactly through their maximal minors and rank-drop loci.

Known reductions and cases. It is enough to study compressed Gorenstein algebras of odd socle degree. The first formerly open Hilbert function

$$(1,3,6,6,3,1)$$

has been settled affirmatively. More recent work proves the weak Lefschetz property when the Sperner number is at most the socle degree plus one; in codimension three, the remaining unknown regime has both socle degree and Sperner number greater than six.

A concrete computational target is to sample or construct ternary dual generators $F$, form the catalecticant and multiplication matrices symbolically, and eliminate $a,b,c$ from the maximal-minor conditions. This could either produce a genuine counterexample or reveal structural identities forcing a Lefschetz element.