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

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 ${\text{PP}}^r$. 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.