Irreducibility of the Hilbert scheme of smooth curves in $\Bbb P^4$ of degree $g+2$ and genus $g$

TL;DR

This paper proves that the Hilbert scheme of smooth, non-degenerate curves of degree g+2 and genus g in P^4 is irreducible for all g, extending previous results that covered only low genus cases.

Contribution

It establishes the irreducibility of the Hilbert scheme al_{g+2,g,4} for all g, removing restrictions on the genus and filling gaps in earlier research.

Findings

The Hilbert scheme al_{g+2,g,4} is irreducible for all g.

Extends previous irreducibility results to all genera g.

Completes the classification of irreducibility for these Hilbert 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 $\PP^r$. In this note, we show that any non-empty $\mathcal{H}_{g+2,g,4}$ is irreducible without any restriction on the genus $g$. Our result augments the irreducibility result obtained earlier by Hristo Iliev(2006), in which several low genus $g\le 10$ cases have been left untreated.