Irreducibility of the Hilbert scheme of smooth curves in $$\mathbb {P}^3$$ of de gree g 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 \(\mathcal {H}_{g,g,3}\) is irreducible without any restriction on the genus g. This extends the result obtained earlier by Iliev (Proc Am Math Soc 134:2823–2832, 2006).