Weak Lefschetz property of equigenerated complete intersections: Applications

Abstract

In this paper, we prove that any Artinian complete intersection homogeneous ideal I in K[x0,··· , xn] generated by n + 1 forms of degree d ≥ 2 satisfies the weak Lefschetz property (WLP) in degree t < d + ⌈d/n ⌉. As a consequence, we get that the Jacobian ideal of a smooth 3-fold of degree d ≥ 6 in P4 satisfies the weak Lefschetz property in degree d, answering a recent question of Beauville [Hyperplane sections of cubic threefolds, Proc. Amer. Math. Soc. 153 (2025), no. 12, 5167–5170]