Weak Lefschetz property of equigenerated complete intersections: Applications

In this paper, we prove that any Artinian complete intersection homogeneous ideal I I in K [ x 0 , ⋯ , x n ] K[x₀,⋯ ,xₙ] 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+ d/n . As a consequence, we get that the Jacobian ideal of a smooth 3-fold of degree d ≥ 6 d≥ 6 in P 4 P⁴ 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].