Consider a projective variety $X$ over the field of the complex numbers, and an action of a reductive group $G$ on $X$ (I will consider the case of $G$ not finite, in particular $G=mathbb{C}^*$). Reading Hu-Keel’s famous paper on Mori dream spaces, Lemma 2.1, I came across the hypothesis of having an action “with finite stabilizer”. Also reading Hausen’s paper “A generalization of Mumford’s Geometric Invariant Theory”, Lemma 5.5, I see the same notion.

*Question:* What is the precise definition of “action with finite stabilizer”? Does it mean for that for every point $pin X$ the isotropy group $G_p$ has finite cardinality?

While this look the more simple solution, I’m a bit confused as having finite stabilizer implies every point is not fixed, and this for example excludes every $mathbb{C}^*$-action on projective spaces for examples. I don’t know, I think this definition is different from the one I’m thinking of; therefore I imagine it’s like common knowledge. I’ve tried to read the proof of these statements but I didn’t find any hint regarding this definition.

I apologize in advance for the simplicity of the question: if you think does not fit the criteria for this forum, I will post it on MSE. I first post it here since it is a question coming from my own research, despite being quite stupid.