Weak Lefschetz property of equigenerated complete intersections. Applications

In this paper, we prove that any Artinian complete intersection homogeneous ideal $I$ in $K[x_0,\cdots ,x_n]$ generated by $n+1$ forms of degree $d\ge 2$ satisfies the weak Lefschetz property (WLP) in degree $t<d+\lceil \frac{d}{n}\rceil$. As a consequence, we get that the Jacobian ideal of a smooth 3-fold of degree $d\ge 7$ in ${\mathbb{P}}^4$ satisfies the weak Lefschetz property in degree $d$, answering a recent question of Beauville.