Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, that isthe union of components whose general point corresponds to a smooth irreducibleand non-degenerate curve of degree $d$ and genus $g$ in $\mathbb P^r$. Acomponent of $\mathcal{H}_{d,g,r}$ is rigid in moduli if its image under thenatural map $\pi:\mathcal{H}_{d,g,r} \dashrightarrow \mathcal{M}_{g}$ is a onepoint set. In this note, we provide a proof of the fact that$\mathcal{H}_{d,g,r}$ has no components rigid in moduli for $g > 0$ and $r=3$.In case $r \geq 4$, we also prove the non-existence of a component of$\mathcal{H}_{d,g,r}$ rigid in moduli in a certain restricted range of $d$,$g>0$ and $r$. In the course of the proofs, we establish the irreducibility of$\mathcal{H}_{d,g,3}$ beyond the range which has been known before.
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