Let $g\ge 0$, $d\ge 1$ be integers. We consider the Moduli space $H_{g,d}$ parametrising smooth irreducible closed curves $C\subset \mathbb{P}^3$ of degree $d$ and genus $g$. Let us denote by $U_{g,d}$ the subset consisting of curves $C$ such that $\{f\in \mathrm{Aut}(\mathbb{P}^3)\mid f(C)=C\}$ is trivial.
For which $g,d$ is $U_{g,d}$ a dense open subset of $H_{g,d}$ ?
I have the impression that $U_{g,d}$ is always open in $H_{g,d}$. I would for this consider $X=\{(f,C)\mid f\in \mathrm{Aut}(\mathbb{P}^3, C\in H_{g,d}\mid f(C)=C\}$, and look at the projection to $H_{g,d}$. The fibres with at least two elements should be closed I guess (unless the ones with one element in the fibre come with multiplicity...).
For $g=0$ and $d\in \{1,2,3\}$, then $U_{g,d}$ is simply empty, as all automorphisms of lines, conics and twisted cubics extend to $\mathbb{P}^3$. Similarly, $U_{1,3}$ is also empty as plane cubics always have some automorphisms extending (finitely many). Is $U_{g,d}$ a dense open subset of $H_{g,d}$ for the others $(g,d)$?
Edit: As pointed by Robert Bryant, all curves contained in a plane gives $U_{g,d}=\varnothing$. One should then forget the cases where $g=\frac{(d-1)(d-2)}{2}$, so $(0,1),(0,2),(1,3),(3,4), (6,5), (10,6),\ldots$