Perazzo \(n\)-folds and the weak Lefschetz property (Q6623827)

A degree \(d\) \textit{Perazzo form} is a homogeneous polynomial \(f \in k[x_0, \dots, x_n, u_1, \dots, u_m]\) of the form\N\[\Nf = \sum_{i=0}^n p_i (u_1, \dots, u_m) x_i + g(u_1, \dots, u_m),\N\]\Nwith \(n \geq m \geq 2\), where \(p_0, \dots, p_n \in k[u_1, \dots, u_m]\) are algebraically dependent but linearly independent forms of degree \(d-1\) and \(g(u_1, \dots, u_m)\) is a form of degree \(d\). Associated to \(f\) is the artinian graded Gorenstein algebra \(A_f \cong R/\mathrm{Ann}_R f\), where \(R\) is the ring of differential operators and the \(\mathrm{Ann}_R f\) is the annihilator of \(f\). Since the Hessian of \(f\) vanishes, \(A_f\) fails the strong Lefschetz property. The Hilbert function is determined by its \(h\)-vector \((h_0, \dots h_d)\) where \(d\) is the socle degree of \(A_f\). The \(h\)-vector is symmetric in the sense that \(h_i = h_{d-i}\) for each \(i\).\N\NFor fixed \(m, n\) and \(d\), the authors show that the \(h\)-vectors of Perazzo algebras \(A_f\) have a maximum \(h_{max}\) and minimum \(h_{min}\) under the obvious partial ordering, generalizing the result for \(m=2\) due to \textit{R. M. Miró-Roig} and \textit{J. Pérez} [J. Algebra 646, 357--375 (2024; Zbl 1539.13049)]. On the other hand, the \(h\)-vectors are not necessarily unimodal as they are when \(m=2\): the authors determine the precise values of \(n \geq m \geq 3\) for which \(h_{max}\) or \(h_{min}\) are not unimodal. In particular, \(h_{max}\) is never unimodal for \(d \gg 0\). The authors prove that Perrazo algebras with \(h\)-vector \(h_{max}\) never satisfy the weak Lefschetz property (WLP), while those with \(h\)-vector \(h_{min}\) unimodal and a minor technical condition do satisfy the WLP. For intermediate \(h\)-vectors, the question of the WLP remains open. An inductive argument is given to calculate the minimal free resolution of a class of Perazzo algebras in \(5\) variables with minimal \(h\)-vector.