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People attribute the following theorem to Weil:

Any variety $X$ equipped with a birational action of a connected algebraic group $G$ is equivariantly birationally isomorphic to a variety $Y$ equipped with a regular action of $G$.

For example, see Remark 2.3.7 in the book "Michel Brion, Preena Samuel, and V. Uma. Lectures on the structure of algebraic groups and geometric applications".

Is there a precise reference for this result - ideally written in the modern language of algebraic geometry?

In my case, the variety $X$ is normal and quasi-projective. Can $Y$ still be normal and quasi-projective?

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    $\begingroup$ Theorem 2.5 in S. Cantat. Morphisms between Cremona groups, and characterization of rational varieties. Compos. Math., 150(7):1107 should be useful. $\endgroup$
    – Ariyan Javanpeykar
    Commented Aug 28 at 11:36
  • $\begingroup$ Thank you! In Cantat's paper, he needs $G$ to have a bounded degree to do the regularization. Since my $G$ is connected, the degree of each element of $G$ should be bounded, right? $\endgroup$
    – Li Yutong
    Commented Aug 28 at 12:26
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    $\begingroup$ I just remembered another useful reference: see arxiv.org/abs/1204.1799 $\endgroup$
    – Ariyan Javanpeykar
    Commented Aug 28 at 16:50
  • $\begingroup$ Bounded degree is indeed automatic, because your G is an algebraic group. Also, have a look at [25] and [35] cited in Cantat's paper. $\endgroup$
    – Ariyan Javanpeykar
    Commented Aug 28 at 16:53
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    $\begingroup$ That is explained completely in “N’eron models” by Bosch, Luetkebohmert, and Raynard. $\endgroup$
    – Jason Starr
    Commented Aug 28 at 19:07

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Besides the nice references mentioned by Ariyan JavanPeykar and Jason Starr, I feel that the paper "Regularization of Rational Group Actions" by Hanspeter Kraft is just what I need (see Section 1.7 and 1.8).

The rough idea is the following: first, by Weil's theorem, there exists a $G$-variety $X'$ which is $G$-equivariantly birational to $X$. Replacing $X'$ by its normalization which is still a $G$-variety. Then by the result of Sumihiro/Brion, there exists an $G$-invariant open set $U \subset X'$ which is quasi-projective. By another result of Sumihiro/Brion, $U$ admits a $G$-equivariant embedding into a projective $G$-variety $Y'$. Finally, take a $G$-equivariant resolution $Y \to Y'$, we get the desired projective and normal variety.

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