In this paper, we prove that any Artinian complete intersection homogeneous ideal I I in K x 0, ⋯, x n Kx₀, , x₍ generated by n + 1 n+1 forms of degree d ≥ 2 d 2 satisfies the weak Lefschetz property (WLP) in degree t > d + ⌈ d n ⌉ t> d+ dn. As a consequence, we get that the Jacobian ideal of a smooth 3-fold of degree d ≥ 6 d 6 in P 4 P^4 satisfies the weak Lefschetz property in degree d d, answering a recent question of Beauville Hyperplane sections of cubic threefolds, Proc. Amer. Math. Soc. 153 (2025), no. 12, 5167–5170.
Beorchia et al. (Wed,) studied this question.