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

TL;DR

This paper investigates the structure and existence of Hilbert schemes of linearly normal curves in projective space with small index of speciality, revealing new existence results and irreducibility properties for specific cases.

Contribution

It establishes the existence and non-existence of Hilbert schemes with index of speciality 4 and determines irreducibility for certain families in a range of dimensions.

Findings

Existence of Hilbert schemes with index of speciality 4.

Non-existence in some sporadic cases.

Irreducibility of specific Hilbert schemes for 3 ≤ r ≤ 8.

Abstract

We study the Hilbert scheme $\mathcal{H}^\mathcal{L}_{d,g,r}$ 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 $\mathcal{H}^\mathcal{L}_{2r+4,r+8,r}$ for $3\le r\le 8$, which are rather peculiar families in a certain sense.