On the Hilbert scheme of linearly normal curves in ${\mathbb{P}}^r$ with small index of speciality

Abstract

We study the Hilbert scheme ${\mathcal{H}}_{d,g,r}^{\mathcal{L}}$ parametrizing smooth, irreducible, non-degenerate and linearly normal curves of degree $d$ and genus $g$ in ${\mathbb{P}}^r$ whose complete and very ample hyperplane linear series $\mathcal{D}$ have relatively small index of speciality $i(\mathcal{D})=g-d+r$. In particular we show the existence (and non-existence as well in some sporadic cases) of every Hilbert scheme of linearly normal curves with $i(\mathcal{D})=4$. We also determine the irreducibility of ${\mathcal{H}}_{2r+4,r+8,r}^{\mathcal{L}}$ for $3\le r\le 8$, which are rather peculiar families in a certain sense.

arXiv comment26 pages. Several typos and grammatical errors have been fixed. Final version to appear in Indagationes Mathematicae