2
$\begingroup$

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?

$\endgroup$
  • 3
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.