Harbourne, Schenck and Seceleanu's Conjecture

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.

Journal of Algebra, to appear