Denote by $$\mathcal {H}_{d,g,r}$$ H d , g , r the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree d and genus g in $$\mathbb {P}^r$$ P r . A component of $$\mathcal {H}_{d,g,r}$$ H d , g , r is rigid in moduli if its image under the natural map $$\pi :\mathcal {H}_{d,g,r} \dashrightarrow \mathcal {M}_{g}$$ π : H d , g , r ⤏ M g is a one point set. In this note, we provide a proof of the fact that $$\mathcal {H}_{d,g,r}$$ H d , g , r has no components rigid in moduli for $$g > 0$$ g > 0 and $$r=3$$ r = 3 , from which it follows that the only smooth projective curves embedded in $$\mathbb {P}^3$$ P 3 whose only deformations are given by projective transformations are the twisted cubic curves. In case $$r \ge 4$$ r ≥ 4 , we also prove the non-existence of a component of $$\mathcal {H}_{d,g,r}$$ H d , g , r rigid in moduli in a certain restricted range of d, $$g>0$$ g > 0 and r. In the course of the proofs, we establish the irreducibility of $$\mathcal {H}_{d,g,3}$$ H d , g , 3 beyond the range which has been known before.