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

## TL;DR

This paper proves the irreducibility of certain Hilbert schemes of linearly normal curves in projective space for a broad range of degrees and genera, extending classical results beyond the Brill-Noether range.

## Contribution

It establishes the irreducibility of the Hilbert scheme components of linearly normal curves for high degrees, extending Severi's conjecture beyond traditional bounds.

## Key findings

- Irreducibility of _{d,g,r} for d  g+r-3 in many cases
- Extension of Severi's conjecture on Hilbert scheme irreducibility
- Validates a modified Severi assertion for specific degree and genus ranges

## 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 $\mathcal{H}^\mathcal{L}_{d,g,r}$ the union of those components of $\mathcal{H}_{d,g,r}$ whose general element is linearly normal and we show that any non-empty $\mathcal{H}^\mathcal{L}_{d,g,r}$ ($d\ge g+r-3$) 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 $\mathcal{H}^\mathcal{L}_{d,g,r}$ of linearly normal curves for $g+r-3\le d\le g+r$, $r\ge 3$, and $g \ge 2r+3$ if $d=g+r-3$.