Ordinary algebraic curves and associated webs. (Q432689)

Given an algebraic curve $\Gamma$ of degree $d$ in $\mathbb P^n$, the $d$ points of a hyperplane section $\Gamma \cap H$ of $\Gamma$ can be interpreted as $d$ hyperplanes in the dual space $\check\mathbb P^n$, and therefore define a $d$-web in $\check\mathbb P^n$. An integral non-degenerate curve (with $d\geq n$ and $n\geq 3$) is called ordinary if its general hyperplane section is of maximal rank, i.e., the natural restriction map $H^0(\mathcal O_H(h))\rightarrow H^0(\mathcal O_{\Gamma\cap H}(h))$ is of maximal rank for all $h$. In this case the minimum degree of a hypersurface of $H$ containing $\Gamma\cap H$, denoted by $k_0+1$, is determined. In the article [\textit{V. Cavalier} and \textit{D. Lehmann}, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 11, No. 1, 197--214 (2012; Zbl 1244.53014)], the notion of ordinary $d$-web of codimension $1$ on a holomorphic manifold of dimension $n$ has been introduced. In the article under review it is proved that, if $k_0\geq 2$, the ordinary curves are precisely those whose associated $d$-web is ordinary.NEWLINENEWLINEThe authors prove that, for all $d\geq 3$, the family of ordinary and arithmetically Cohen-Macaulay curves of degree $d$ in $\mathbb P^3$ is non-empty and fills an irreducible component of the appropriate Hilbert scheme. They also show how to construct explicitly these curves via their minimal free resolution. The property of being ordinary is studied for rational singular curves in $\mathbb P^3$ and for complete intersections, thus providing examples and counterexamples.

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