Let $X$ be a compact complex protective surface that admits a nontirvial algebraic $\mathbb C^\*$-action. It seems to me, that the ample cone of $X$ is polyhedral with finite number of faces. I wonder if this is statement is correct and whether it is written down in some book/article.



The ample cone of a surface with an algebraic $\mathbb C^*$ action
This is true, and follows from Corollary 3.29 of <https://arxiv.org/pdf/0811.0517.pdf>


