On the Hilbert scheme of linearly normal curves in ${\mathbb{P}}^r$ of relatively high degree

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 $\PP^r$. 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 and we show that any non-empty ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ ($d\ge g+r-3$) is irreducible for an extensive range of triples $(d,g,r)$ beyond the Brill-Noether range. This establishes the validity of a suitably modified assertion of Severi regarding the irreducibility of the Hilbert scheme ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ of linearly normal curves for $g+r-3\le d\le g+r$, $r\ge 3$, and $g\ge 2r+3$ if $d=g+r-3$.

Introductory section has been rewritten together with a historical remark concerning Severi's assertion on the Hilbert scheme of smooth curves. Some of main results have been stated precisely, especially on the existence part. Several typos corrected and references updated with some expository improvement. Final version. arXiv admin note: substantial text overlap with arXiv:1903.02307