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 = ( ℓ 1 2 , … , l r + 1 2 ) ⊂ K [ x 1 , … , x r ] generated by the square of r + 1 general linear forms ℓ 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 [5] , Theorem 6.1. [ABSTRACT FROM AUTHOR]