# ag.algebraic geometry – Actions with finite stabilizer

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.