Harbourne, Schenck and Seceleanu's Conjecture

TL;DR

This paper proves Harbourne, Schenck, and Seceleanu's conjecture that certain artinian ideals generated by squares of general linear forms fail the Weak Lefschetz property for specific numbers of variables, completing the proof for all cases.

Contribution

The paper confirms the conjecture for all relevant cases, extending previous partial results and providing a complete proof.

Findings

The ideal fails the Weak Lefschetz property for r=6.

The ideal fails the Weak Lefschetz property for all r ≥ 8.

Partial cases for even r were previously proved; this work completes the proof.

Abstract

In [HSS], Conjecture 5.5.2, Harbourne, Schenck and Seceleanu conjectured that, for $r=6$ and all $r\ge 8$, the artinian ideal $I=(\ell _1^2,\dots ,l_{r+1}^2)\subset K[x_1, \dots ,x_r]$ generated by the square of $r+1$ general linear forms $\ell _{i}$ fails the Weak Lefschetz property. This paper is entirely devoted to prove this Conjecture. It is worthwhile to point out that half of the Conjecture - namely, the case when the number of variables $r$ is even - was already proved in [mmn], Theorem 6.1.