Exceptional algebraic curves for infinite groups of $\text{PGL}(n,\mathbb{C})$

We classify algebraic curves in ${\mathbb{C}\mathbb{P}}^n$ ($n\ge 2$) that are invariant under an infinite subgroup of $\mathrm{PGL}(n+1,\mathbb{C})$. In particular, we prove that any irreducible, non-degenerate, one-dimensional algebraic set in ${\mathbb{C}\mathbb{P}}^n$ invariant under an infinite subgroup of $\mathrm{Aut}({\mathbb{C}\mathbb{P}}^n)$ must be projectively equivalent to a monomial curve.