Ordinary curves, webs and the ubiquity of the weak Lefschetz property
A standard graded artinian algebra \(R/I\), where \(R\) is a polynomial ring over a field, is said to have the \textit{Weak Lefschetz property} (WLP) if there exists a linear form \(L\) for which the induced multiplication from any component of \(R/I\) to the next has maximal rank. This property has been intensely studied over the last decade or so. Many natural algebras have been shown to have this property, and connections to many seemingly unrelated topics have been found. This paper provides another such example. Let \(C \subset \mathbb P^n\) be an integral, non-degenerate curve and let \(\Gamma\) be its intersection with a general hyperplane \(H\). Then \(C\) is said to be \textit{ordinary} if \(\Gamma\) has generic Hilbert function: \(h^0(H, \mathcal I_\Gamma (t)) \cdot h^1 (H, \mathcal I_\Gamma(t)) = 0\) for all \(t\). The \textit{postulation} of \(C\) is defined as \(\displaystyle s(C) = \min \{ \ell \in \mathbb Z \;| \;H^0(\mathbb P^n, \mathcal I_C (\ell)) \neq 0 \}\) and the \textit{index of speciality} of \(C\) is \(\displaystyle e(C) = \max \{ \ell \in \mathbb Z \;| \;H^2 (\mathbb P^n, \mathcal I_C (\ell)) \neq 0 \}\). The \textit{Rao module} of \(C\) is the finite length module \(\displaystyle M(C) = \bigoplus_{\ell \in \mathbb Z} H^1 (\mathbb P^n, \mathcal I_C (\ell))\). Finally, we have the notation \(\displaystyle \sigma(C) = \min \{ \ell \in \mathbb Z \;| \;H^0(H,\mathcal I_{C \cap H}(\ell)) \neq 0 \}\). The main result of the paper is the following. Assume that \(M(C)\) is cyclic, that \(M(C)\) has the WLP, and that \(e(C)+1 < \sigma(C) < s(C)\). Then \(C\) is an ordinary curve.
scientific article; zbMATH DE number 6370052
scientific article