Weak Lefschetz property of equigenerated complete intersections. Applications

TL;DR

This paper establishes a sharp range for the weak Lefschetz property (WLP) for equigenerated complete intersections in $R=K[x_0,\dots,x_n]$ by leveraging the associated stable syzygy bundle $\mathcal{E}$ and applying Grauert-Mülich and Flenner restriction theorems. The main result shows that if $I=(f_0,\dots,f_n)$ with $\deg f_i = d\ge2$, then the WLP holds in all degrees $t< d+\left\lceil \frac{d}{n} \right\rceil$, with a parallel general statement for arbitrary degrees via bundle-splitting considerations. These bounds yield new WLP results for Jacobian ideals of smooth hypersurfaces, giving WLP in degrees $t< d-1+\left\lceil \frac{d-1}{n} \right\rceil$, and imply geometric consequences such as the generic finiteness of hyperplane-section variation in high-degree settings. In particular, for a smooth 3-fold of degree $d\ge 7$ in ${\mathbb P}^4$, the Jacobian ideal has WLP in degree $d$, answering Beauville’s question in that case and extending known results in codimension two and three.

Abstract

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.

Theorems & Definitions (24)

• Definition 2.1
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Theorem 2.4
• proof
• Theorem 3.1
• proof
• Remark 3.2