Let $G$ be algebraic group acting on a smooth algebraic variety $M$. Assume the action is proper and free. Is the orbit space $M/G$ an algebraic variety? If so, could someone point to a reference? If not, could someone point to a counterexample? Thanks.



Quotient of smooth algebraic variety by proper free action of algebraic group
The answer is **no**.


A counterexample is given in the paper by James K. Deveney and David R. Finston


[A Proper $G\_a$ Action on $\mathbb{C}^5$ Which is Not Locally Trivial](http://www.jstor.org/stable/2160782),
Proc. Amer. math. Soc. **123** (1995).


Quoting from the introduction:



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> An assertion which would imply that any proper, fixed point free $G\_a$-action on a normal variety is locally trivial and admits a quasi-projective quotient appears in a paper of Magid and Fauntleroy [5], and the source of the error is pointed out in [4]. The example here indicates that no such result is possible.
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