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?

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    $\begingroup$ 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 $\endgroup$ – abx Aug 26 at 16:18

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