Let $C$ and $D$ be two projective plane curves (over $\mathbb{C}$) of degree $d>1$. Suppose that $C$ and $D$ are isomorphic.
Are $C$ and $D$ projectively equivalent?
For smooth curves this is a theorem by M. Noether. So we may restrict to the singular curves.
Any reference, idea or counterexample will be welcome.
Using Hartshorne's generalization of Noether's Theorem - Theorem 2.1 here - we have that a generalized divisor $L$ on a plane curve $C$ of degree at least $4$ satisfy $h^0(C,L)=3$ if and only if $L$ is given by a hyperplane section.
Then we follow the same argument of this MO answer to conclude that an isomorphism $\phi: C \to D$ has to be the restriction of an element of $\mbox{PGL}(3,\mathbb{C})$.
It remains to prove in low degrees. This can be done by describing all possible singular curves. However the assertion is false for degree three smooth curves.