# Weak Lefschetz property Jacobian ring smooth hypersurface

Let $$A_{.}$$ be a graded commutative ring. We say that $$A_{.}$$ satisfies the weak Lefschetz property if for generic $$L \in A_1$$ the multiplication maps $$\times L : A_i \longrightarrow A_{i+1}$$ has maximal rank (analogy with the weak Lefschetz theorem for the cohomology of smooth projective varieties).

I am aware of some work of on this subject (Migliore, Nagel, Hausel, Watanabe, Dimca...), but I can't find a reference for the Jacobian ring of a smooth hypersurface in $$\mathbb{P}^n_{\mathbb{C}}$$.

Is it known if the Jacobian ring of a smooth projective hypersurface satisfies the weak Lefschetz property? If so, is it a consequence of Griffiths residue computations and the weak Lefschetz theorem for the comohology of the hypersurface?

• This is conjectured (more generally, for any graded Artinian complete intersection ring), but not proved as far as I know. This paper from last year gives some partial results, but far from a complete answer – abx Aug 26 at 16:18