# On the Hilbert scheme of linearly normal curves in $\mathbb{P}^4$ of degree $d = g+1$ and genus $g$

## TL;DR

This paper investigates the structure of the Hilbert scheme of linearly normal curves in projective 4-space with degree one more than their genus, showing irreducibility except for a special case where it splits into two components.

## Contribution

It proves the irreducibility of the Hilbert scheme for most cases and describes the reducibility when the genus is 9, confirming a modified Severi conjecture.

## Key findings

- Most Hilbert schemes are irreducible with a single component.
- For genus 9, the scheme has two components, each with linearly normal general elements.
- The results support a modified Severi conjecture on the irreducibility of these schemes.

## Abstract

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb{P}^r$. In this article, we show that any non-empty $\mathcal{H}_{g+1,g,4}$ has only one component whose general element is linear normal unless $g=9$. If $g=9$, we show that $\mathcal{H}_{g+1,g,4}$ is reducible with two components and a general element of each component is linearly normal. This establishes the validity of a certain modified version of an assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ for the case $d=g+1$ and $r=4$.