On the weak Lefschetz property for ideals generated by powers of general linear forms

TL;DR

This work studies the weak Lefschetz property for algebras defined by powers of general linear forms, focusing on I_{n,a} = (x_1^a,...,x_n^a,(x_1+...+x_n)^a). By analyzing initial ideals in reverse lexicographic order and applying Wiebe’s transfer, the authors derive sharp injectivity bounds for multiplication by a general linear form in degree d: n ≥ 3d-2 for the square case (a=2) and n ≥ (3d-3)/2 for the cube case (a=3); they also prove failure of injectivity for squares when n < 3d-2, establishing sharpness. The approach hinges on a detailed monomial description of initial ideals (including D_k sets for squares) and recurrence relations for Hilbert function data, together with connections to Fröberg’s conjecture up to certain degrees. The results yield concrete thresholds for WLP in these almost complete intersections and relate the injectivity phenomenon to combinatorial lattice-path conjectures in related settings.

Abstract

We provide a description of initial ideals for almost complete intersections generated by powers of general linear forms and prove that WLP in a fixed degree $d$ holds when the number of variables $n$ is sufficiently large compared to $d$. In particular, we show that if $n\geq 3d-2$ then WLP holds for the ideal generated by squares at the degree $d$ spot and for $n\ge \frac{3d-3}{2}$ WLP holds for ideal generated by cubes at the degree $d$ spot. Finally, we prove that WLP fails for the ideal generated by squares when $n< 3d -2$ at the $d$th spot by finding an explicit element in the kernel of the multiplication by a general linear form. This shows that our bound on $n$ is sharp in the case of the squares.

Theorems & Definitions (48)

• Theorem 1.1
• proof
• Remark 1.3
• Proposition 1.4: Wiebe, Proposition 2.9
• Lemma 1.5: Eddings-Vraciu, Lemma 3.1
• Theorem 1.6: Rei-Rob-Roi, Theorem 1
• Remark 1.8
• Theorem 2.1
• proof
• Theorem 2.2