Quotients of schemes by connected groups

Question

Let $X$ be a variety over $k$ where the characteristic of $k$ is zero. Let $G$ be a connected reductive group scheme acting freely and properly on $X$.

By the Keel-Mori theorem, the quotient $X/G$ is represented by an algebraic space. I would like to know if this can always be represented by a scheme, or else to construct a counterexample.

Let me remark that the standard counterexample to this question involves a finite group acting freely and properly on Hironaka’s threefold. I insist on the group $G$ to be connected

Answer 1

$\DeclareMathOperator\PGL{PGL}\DeclareMathOperator\GL{GL}$It seems that J. Kollár gave some examples in his paper "Non-quasi-projective moduli spaces".

In pg. 1080, I think that there is an example of a proper free action of $\PGL(3)$ on a smooth quasi-projective variety $U$ such that the quotient $U/{\PGL(3)}$ is not a scheme. To be precise, $U$ is an open subscheme of the (projectivized) linear series of curves $\lvert\mathcal{O}_{\mathbb{P}^2}(d)\rvert$ and $\PGL(3)$ acts as automorphisms of $\mathbb{P}^2$. We have that $U$ is the open subset parametrizing curves $C$ such that:

(1) $C$ is integral, and the geometric genus of the normalization is $> \frac{1}{2}\binom{d-1}{2}$. (In the paper it says smooth irreducible instead of integral, but I think that this might be a typo?)

(2) The stabilizer $C$ under the action of $\PGL(3)$ is trivial.

Then it is show that the action on $U$ is free and proper (for properness of the action one may use Stacks Lemma 32.16.2, which works for algebraic spaces with quasicompact separated diagonal, to assume that the generic fiber of the family over the DVR is smooth, as he does in his Claim 4). Kollár shows that for $d$ large, the quotient is a smooth algebraic space which is not a scheme.

I believe that the same would hold for the action of $GL(3)$ on the corresponding open subset $\tilde{U}$ of the affine (non-projectivized) space $H^0(\mathbb{P}^2, \mathcal{O}_{\mathbb{P}^2}(d))$. Then the corresponding étale $\GL(3)$-torsor on the quotient $\tilde{U}/{\GL(3)}$ should not be Zariski locally trivializable.

Answer 2

Edit. As user @Johan observes, I need to find a reference for the following: every $G$-torsor for a "special" algebraic group $G$ (thus Nisnevich locally trivial) is Zariski locally trivial. This is true for $G$-torsors over schemes, but the argument below needs this for algebraic spaces.

Original post. I am posting my comments 1 2 3 as an answer, mostly to address the issue raised by user @Johan 1 2. There is indeed some concern if you allow $X$ to be very singular. However, in the case that $X$ is normal so that also $X/G$ is normal, Nisnevich's theorem (solution of the Grothendieck–Serre conjecture over spectra of DVRs) produces sections of the $G$-torsor $X\to X/G$ away from codimension $2$. Since this $G$-torsor is an affine morphism, these sections should extend over codimension $2$ points by S2 extension (I guess I am not sure of a reference for that for algebraic spaces, but since extensions are unique if they exist, it should follow from the usual reference in EGA together with étale descent).

So if there are examples where a free quotient by a reductive group is an algebraic space that is not a scheme, then there are such examples where the group is a semisimple group with no factors that are "special" in the sense of Serre, i.e., no $A_n$-factors or $C_n$-factors.