Isomorphic plane curves and projective equivalence

Question

1) If $C\subset\mathbb{P}^2$ is a smooth degree $d$ plane curve, does there exist another degree $d$ curve $C'\subset\mathbb{P}^2$ such that $C\cong C'$, but $C$ and $C'$ are not related by a $PGL_3$ action?

2) If so, does there exist a 1-dimensional family of such $C'$? (i.e. is the space of plane curves of degree $d$ isomorphic to $C$ of dimension at least 8+1=9?)

(My guess is that the answer to 1) is no for $C$ general, but yes for specific $C$ and the answer to 2) is no for all $C$. However, I feel like there should be a reference, and there's no need to guess.)

Answer 1

If $C\subset\mathbb{P}^2$ is a smooth degree $d$ plane curve, does there exist another degree $d$ curve $C'\subset\mathbb{P}^2$ such that $C\cong C'$, but $C$ and $C'$ are not related by a $\text{PGL}_3$ action?

Actually the answer is no for all smooth plane curves. In fact, the only line bundle on a smooth plane curve of degree $d$ with $3$ independent sections is the hyperplane bundle. This is a consequence of an old theorem of Max Noether, but there is a gap in the proof. There is a modern account in a paper of Ciliberto (Sem. di Geometria, Univ. di Bologna (1982-83), 17-43) and a version that covers singular curves as well in a paper of Hartshorne (J. Math. Kyoto Univ. 26-3 (1986), 375-386).

Answer 2

What you ask is equivalent to finding a very ample line bundle $L$ of degree $d$ on a smooth curve $C$ with $h^0(L)=3$ and if two such are not projectively equivalent, then the corresponding line bundles are different. Since $d\leq 3$ is well understood, let me assume that $d\geq 4$. Then, $L^{d-3}=K_C$ and thus two such line bundles will differ by a torsion line bundle of order $d-3$. There are only finitely many such line bundles and thus there are only finitely many such embeddings. This at least answers your part 2.