Lefschetz properties of Jacobian algebras and Jacobian modules

Abstract

Let $V:f=0$ be a hypersurface of degree $d\ge 3$ in the complex projective space ${\mathbb{P}}^n$, $n\ge 3$, having only isolated singularities. Let $M(f)$ be the associated Jacobian algebra and $H:\ell =0$ be a hyperplane in ${\mathbb{P}}^n$ avoiding the singularities of $V$, but such that $V\cap H$ is singular. We related the Lefschetz type properties of the linear maps $\ell :M(f{)}_k\to M(f{)}_{k+1}$ induced by the multiplication by linear form $\ell$ to the singularities of the hyperplane section $V\cap H$. Similar results are obtained for the Jacobian module $N(f)$.

arXiv commentThis new version, where some details are added and some references are updated, is accepted for publication in Annali della Scuola Normale Superiore di Pisa