Jacobian ideals, arrangements and the Lefschetz properties
The author studies the so-called Weak and Strong Lefschetz Properties. Let \(R = \mathbb{C}[x_{0},\dots,x_{n}]\) be the graded polynomial ring in \(n+1\) variables over \(\mathbb{C}\) and \(I\subseteq R\) an ideal such that \(A = R / I = \bigoplus_{i=0}^{s}A_{i}\) is a graded artinian ring. If there exists a linear form \(L\) such that the multiplication maps \(m_{L,i} : A_{i} \rightarrow A_{i+1}\) have maximal rank for every degree \(i\), then one says that \(A\) has the Weak Lefschetz Property (WLP for brevity). If there exists a linear form \(L\) such that the multiplication maps \(m_{L^{k},i}: A_{i} \rightarrow A_{i+k}\) have maximal rank in every degree \(i\) and for every \(k\), then one says that \(A\) has the Strong Lefschetz Property (SLP for brevity). Let us recall that for a hypersurface \(f=0\) in \(\mathbb{P}^{n}\) we denote by \(J(X)\) its Jacobian ideal, i.e., \(J(X) = (\partial f / \partial x_{0},\dots,\partial f / \partial x_{n})\). Now we are ready to present the main results of the paper. Proposition 1. Let \(X : f= 0\) be a hypersurface in \(\mathbb{P}^{n}\) of degree \(d>2\) such that its singular locus \(X_{s}\) has codimension at most \(n-3\), then the ideal \(J(X)\) has the WLP in degree \(d-2\). Proposition 2. Let \(X : f=0\) be a hypersurface in \(\mathbb{P}^{n}\) of degree \(d > 2\) such that its singular locus \(X_{s}\) has dimension at most \(n-3\), then for every positive integer \(k < d-1\) the ideal \(J(X)\) has the SLP in degree \(d-k-1\) at range \(k\). Proposition 3. Let \(X \subset \mathbb{P}^{n}\) be a general hypersurface and \(J(X)\) its Jacobian ideal, then \(J(X)\) has the SLP. In the rest part of the paper, the author presents some new examples of singular hypersurfaces providing failures of the SLP. It is worth emphasizing that these examples are related to the well-known reflection line arrangements, and we can join them additionally with a quite new topic of the so-called unexpected curves, which were studied recently by \textit{D. Cook II} et al. [``Line arrangements and configurations of points with an unusual geometric property'', Preprint, \url{arXiv:1602.02300}].
scientific article; zbMATH DE number 6890937
scientific article