Irreducibility of the Hilbert scheme of smooth curves in $$\mathbb {P}^4$$ of de gree $$ g+2$$ 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 \(\mathbb {P}^r\). In this note, we show that any non-empty \(\mathcal {H}_{g+2,g,4}\) is irreducible, generically smooth, and has the expected dimension \(4g+11\) without any restriction on the genus g. Our result augments the irreducibility result obtained earlier by Iliev (Proc Am Math Soc 134:2823–2832, 2006), in which several low genus \(g\le 10\) cases have been left untreated.