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We characterize algebraic curves in $\mathbb{P}^{n-1}_\mathbb{C}$ invariantunder an infinite subgroup of $\textrm{PGL}(n,\mathbb{C})$, $n\geq 3$. Moreprecisely, it is shown that given a one-dimensional algebraic set in$\mathbb{P}^{n-1}_\mathbb{C}$ which is invariant under an infinite subgroup ofprojective transformations should satisfy that each irreducible non-degeneratecomponent is projectively equivalent to a monomial curve.

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