Harbourne, Schenck and Seceleanu's Conjecture

In [2], Conjecture 5.5.2, Harbourne, Schenck and Seceleanu conjectured that, for r=6 and all r≥8, the artinian ideal I=(ℓ12,…,lr+12)⊂K[x1,…,xr] generated by the square of r+1 general linear forms ℓi 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 [5], Theorem 6.1.