Existence and the reducibility of the Hilbert scheme of linearly normal curves in ${\mathbb{P}}^r$ of relatively high degrees

Abstract

Let ${\mathcal{H}}_{d,g,r}$ be the Hilbert scheme parametrizing smooth irreducible and non-degenerate curves of degree $d$ and genus $g$ in ${\mathbb{P}}^r.$ We denote by ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ the union of those components of ${\mathcal{H}}_{d,g,r}$ whose general element is linearly normal. In this article we show that ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ ($d\ge g+r-3$) is non-empty in a certain optimal range of triples $(d,g,r)$ and is empty outside the range. This settles the existence (or non-emptiness if one prefers) of the Hilbert scheme ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ of linearly normal curves of degree $d$ and genus $g$ in ${\mathbb{P}}^r$ for $g+r-3\le d\le g+r$, $r\ge 3$. We also determine all the triples $(d,g,r)$ with $g+r-3\le d\le g+r$ for which ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ is reducible (or irreducible).

arXiv comment26 pages. After some feedbacks from the referees, several changes in style have been made to avoid unnecessary misunderstandings. The final section on curves of index of speciality four has been deleted in this final version. The introductory section was rewritten. Table 2 in the last part of the paper was modified with a better explanation. To appear in JPAA