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

TL;DR

This paper investigates the existence and reducibility of the Hilbert scheme of linearly normal curves in projective space, establishing optimal conditions for non-emptiness and classifying cases of reducibility.

Contribution

It determines the precise range of degrees where the Hilbert scheme of linearly normal curves exists and classifies when these schemes are reducible or irreducible.

Findings

Non-emptiness of the Hilbert scheme in the range $g+r-3 \,\leq\, d \leq g+r$ for $r \geq 3.

Complete classification of reducibility versus irreducibility within this range.

Identification of the optimal degree range for existence of linearly normal curves.

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 $\mathcal{H}^\mathcal{L}_{d,g,r}$ the union of those components of $\mathcal{H}_{d,g,r}$ whose general element is linearly normal. In this article we show that $\mathcal{H}^\mathcal{L}_{d,g,r}$ ($d\ge g+r-3$) 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 $\mathcal{H}^\mathcal{L}_{d,g,r}$ 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 $\mathcal{H}^\mathcal{L}_{d,g,r}$ is reducible (or irreducible).