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

Abstract

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