On the Hilbert scheme of smooth curves in $\mathbb{P}^4$ of degree $d = g+1$ and genus $g$ with negative Brill-Noether number

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 article, we show that for low genus $g$ outside the Brill-Noether range, the Hilbert scheme $\mathcal{H}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of $\mathcal{H}_{d,g,r}$ outside the Brill-Nother range for $d=g+1$ and $r=4$.