Rational curves of Castelnuovo-Mumford regularity \(d - r+1\)
A coherent \({\mathcal O}_{{\mathbb P}^r}\)-module \({\mathcal F}\) is called \(n\)-regular if \(H^i({\mathcal F}(n-i))=(0)\) for all \(i>0\). A subscheme \(X\) of \({\mathbb P}^r\) is called \(n\)-regular if its ideal sheaf \({\mathcal I}_X\) is \(n\)-regular. In 1983, \textit{L. Gruson}, \textit{R. Lazarsfeld} and \textit{C. Peskine} [Invent. Math. 72, 491--506 (1983; Zbl 0565.14014)] proved that every (irreducible and reduced) curve of degree \(d\) in \({\mathbb P}^r\) is \((d-r+2)\)-regular and classified the curves which are not \((d-r+1)\)-regular. Contributions to the subsequent problem of classifying the curves which are not \((d-r)\)-regular have been given by Ellia and D'Almeida, for non-hyperelliptic curves of geometric genus \(g\geq 3\), and by Noma. The main theorem of the article under review states that if \(X\) is a rational curve in \({\mathbb P}^r\), with \(r\geq 4\), \(d-r\geq 4\), then \(X\) is not \((d-r)\)-regular if and only if \(X\) has one (and only one) \((d-r+1)\)-secant line. The proof relies on a refinement of the method of Gruson-Lazarsfeld-Peskine. If \(\phi: {\mathbb P}^1 \to {\mathbb P}^r\) is the natural morphism whose image is \(X\), the Author considers the sheaf \(M=\phi^*(\Omega^1_{{\mathbb P}^r}(1))\) and shows that the non \((d-r)\)-regular curves fall in three classes, corresponding to different splitting types of \(M\), which are explicitly described. He also shows that the assumption \(d-r\geq 4\) is necessary, giving examples of \(4\)-regular but non \(3\)-regular curves without any \(4\)-secant line, with \(r=5\) and \(d=8\), over an algebraically closed field of characteristic \(17\).
scientific article; zbMATH DE number 5572158
scientific article