Projective curves, hyperplane sections and associated webs

Abstract

An integral and non-degenerate curve $C\subset \mathbb {P}^r$ is said to be ordinary (Gruson, Hantout and Lehmann) if if the general hyperplane section $H\cap C$ of $H$ is of maximal rank in $H$. Let $g'(r,d)$ be the maximal integer such that for every $g\in \{0,\dots ,g'(r,d)\}$ there is a smooth ordinary curve $C\subset \mathbb {P}^r$ with degree $d$ and genus $g$. Here we discuss the relevance of old papers to get a lower bound for $g'(r,d)$. We prove that arithmetically Gorenstein curves $C \subset \mathbb {P}^r$ are ordinary only if either $r=2$ or $d =r+1$ and $\omega _C \cong \mathcal {O}_C$. We prove that general low genus curves are ordinary.