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Assume that $k$ is an infinite field. Let $G$ be a finite (constant) group, let $V$ be a faithful $G$-representation over $k$, $U$ a non-empty open subset of $V$ where the action is free. The map $\pi:U\to U/G$ is a $G$-torsor, and moreover, every $G$-torsor over $k$ arises as a fiber of $\pi$, and the set of $p\in (U/G)(k)$ such that $\pi^{-1}(p)$ is isomorphic to a given torsor is dense in $U/G$. This is actually true over every field extension $K/k$.

Let $\alpha,\beta$ be two $G$-torsors over $k$, and assume that there exists a $G$-torsor $P\to Z$ where $Z$ is an open subset of $\mathbb{P}^1$, and assume that $\alpha$ and $\beta$ both arise as fibers of $P\to Z$. Can I find a map from some open subset $Z'$ of $\mathbb{P^1}$ to $U/G$ such that the induced torsor above $Z'$ admits $\alpha$ and $\beta$ as fibers? The converse is of course obvious.

Note that if $U/G$ is retract rational (condition which does not depend on $V$) but only on $G$), then any two points on $U/G$ may be connected by an open subset of a rational curve, so everything is trivial in this case. Unfortunately the only examples that I have so far belong to this trivial class.

One might of course ask a similar question for an arbitrary linear algebraic group $G$ (in this case $V$ must be chosen generically free and not just faithful, and the quotient will be a rational quotient).

Added: the same question, but with a chain of rational curves in place of a single one.

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  • $\begingroup$ If you change $P\to Z$ by an automorphism of $Z$, I believe the image remains the same, or I did not understand your suggestion. $\endgroup$ – Scott Foster Oct 1 '18 at 3:20
  • $\begingroup$ If $\rho:P\rightarrow Z$ is the torsor, then I think I can show there is an open $W$ of $Z$ with $\rho^{-1}(W)$ the fiber along a map $W\rightarrow U/G$. It's a bit of a weaker statement though. $\endgroup$ – Eoin Oct 1 '18 at 9:28
  • $\begingroup$ Yes, it's weaker because maybe the two interesting points lie outside of $W$. $\endgroup$ – Scott Foster Oct 1 '18 at 15:43
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Welcome new contributor. That is true, even for $G$ a geometrically reductive group scheme, not merely for finite groups. The proof is elementary, but it is a bit long. It is related to the problems of weak approximation and strong approximation, just for affine space in your original question. Strong approximation is elementary for affine space. However, there are many other schemes that satisfy weak approximation and strong approximation (sometimes by deep arguments), and the analogous result also holds for those schemes.

Let $S$ be a scheme. Let $\pi_S:V_S\to S$ be a smooth morphism, resp. a smooth group $S$-scheme, a smooth simplicial $S$-scheme, etc.

Definition 1. For every $S$-scheme $Z$, a form of $V_S$ over $Z$ is a smooth morphism $\pi:V_Z\to Z$, resp. a smooth $Z$-group scheme, a smooth simplicial $Z$-scheme, etc., such that there exists a smooth, surjective morphism $Z'\to Z$ and an isomorphism of $V_Z\times_Z Z'$ with $V_S\times_S Z'$ as smooth $Z'$-schemes, resp. as smooth group $Z'$-schemes, smooth simplicial $Z'$-schemes, etc.

Definition 2. Let $\mathcal{C}$ be a class of affine $S$-schemes. All forms of $V_S/S$ are said to satisfy strong approximation, resp. weak approximation, with respect to $\mathcal{C}$ if for every affine $S$-scheme $Z$ in the class $\mathcal{C}$, for every form $\pi:V_Z\to Z$ of $V_S/S$, for every closed subscheme $i:\Pi\hookrightarrow Z$ that is Artinian, and for every $Z$-morphism $\sigma_\Pi:\Pi\to V_Z$, there exists a $Z$-morphism $\sigma:Z\to V_Z$ that restricts to $\sigma_\Pi$, resp. after replacing $Z$ by an open subscheme containing $\Pi$, there exists a $Z$-morphism $\sigma$ that restricts to $\sigma_\Pi$.

Forms of $V_S/S$ satisfy very weak approximation with respect to $\mathcal{C}$ if every form of $V_S/S$ over an infinite field has a Zariski dense set of rational points and for every $Z/S$ in $\mathcal{C}$, for every form $V_Z$ of $V_S$ over $Z$, for every Artinian closed subscheme $\Pi$ of $Z$ whose residue fields are infinite, and for every closed subset $W\subset V_Z$ whose fiber $W_p\subset V_{Z,p}$ is nowhere dense for every point $p\in \Pi$, after replacing $Z$ by an open subscheme containing $\Pi$, there exists a $Z$-morphism $\sigma$ such that $\sigma(\Pi)$ is disjoint from $W$.

Lemma 3. If forms of $V_S/S$ satisfy strong approximation, then they satisfy weak approximation. If forms of $V_S/S$ over every infinite field have a Zariski dense set of rational points and if they satisfy weak approximation, then they satisfy very weak approximation.

Proof. The first claim is obvious. For the second claim, use Zariski density of points to find a rational point on each fiber $V_{Z,p}$ that is in the complement of $W_p$. Next, use $Z$-smoothness of $V_Z$ to extend that rational point to a section $\sigma_\Pi$. QED

Let $R$ be an irreducible scheme, and let $Q\subset R$ be a proper closed subset.

Corollary 4. Assume that $V_S/S$ satisfies very weak approximation for class $\mathcal{C}$. For every $Z/S$ in class $\mathcal{C}$, for every form $V_Z/Z$ of $V_S/S$, for every Artinian, closed subscheme $\Pi$ of $Z$ with infinite residue fields, and for every morphism $f:V_Z\to R$ such that $f_p:V_{Z,p}\to R$ is dominant for every $p\in \Pi$, there exists a section $\sigma$ of $\pi$ such that $f\circ \sigma(\Pi)$ is disjoint from $Q$.

Proof. Define $W$ to be the inverse image closed subset $f^{-1}(Q)$. By the hypothesis, for every $p\in \Pi$, the fiber $W_p$ is a proper closed subset of the fiber $V_{Z,p}$. QED

There are many examples of morphisms that satisfy strong approximation, resp. weak approximation for a specified class of affine schemes. One example that satisfies strong approximation for all affine schemes is affine space itself.

Every form of the additive group scheme $\mathbb{A}^r_S$ over $Z$ is a geometric vector bundle $$\pi:V_Z\to Z$$, i.e., $\text{Spec}_Z(\text{Sym}^\bullet_{\mathcal{O}_Z}(\mathcal{E}))$ for a locally free $\mathcal{O}_Z$-module $\mathcal{E}$ of finite rank $r$. Let $$i:\Pi \hookrightarrow Z$$ be a closed subscheme that is Artinian. Let $$\sigma_{\Pi}:\Pi \to V_Z$$ be a $Z$-morphism, i.e., $\pi\circ \sigma_{\Pi}$ equals $i$.

Strong Approximation for Affine Space. There exists a section $\sigma:Z\to V_Z$ of $\pi$ whose restriction to $\Pi$ equals $\sigma_{\Pi}$. Also, rational points are dense on every fiber over an infinite field.

Proof. The fact that $k$-points of $\mathbb{A}^r_k$ are Zariski dense for every infinite field is elementary; the main claim is strong approximation.

The $\mathcal{O}_Z(Z)$-module $\mathcal{E}(Z)$ is locally free of finite rank $r$. Thus, we can check locally that for every $\mathcal{O}_Z(Z)$-module $N$, the adjointness homomorphism, $$\text{Hom}_{\mathcal{O}_Z(Z)}(\mathcal{E}(Z),\mathcal{O}_Z(Z))\otimes_{\mathcal{O}_Z(Z)} N \to text{Hom}_{\mathcal{O}_Z(Z)}(\mathcal{E}(Z),N)$$ is an isomorphism. In particular, setting $N$ equals to $\mathcal{O}_{\Pi}(\Pi)$, it follows that every $\mathcal{O}_Z(Z)$-module homomorphism from $\mathcal{E}(Z)$ to $\mathcal{O}_{\Pi}(\Pi)$ is the restriction of a homomorphism to $\mathcal{O}_Z(Z)$. Now use the universal property of the relative Spec and of the symmetric algebra. QED

Forms Associated to a Torsor. Let $S$ be a scheme. Let $G$ be an affine, flat, finitely presented $S$-group scheme. Let $V_S \to S$ be an affine, smooth $S$-scheme together with an $S$-action of $G$, $$\mu_V:G\times_S V_S \to V_S.$$ For every $S$-scheme $E$, the base change $V_S\times_S E \to E$ is a smooth, affine $E$-scheme. For every action of $G$ on $E$, $$\mu_E:G\times_S E \to E,$$ there is an associated diagonal action, $$(\mu_V,\mu_E):G\times_S (V_S\times_S E) \to V_S\times_S E$$ that is equivariant for both projections.

For every $S$-scheme $Z$ and for every $G$-torsor over $Z$, $$(E\to Z,\mu_E:G\times_S E \to E),$$ by fppf descent for affine morphisms, there is a unique smooth, affine $Z$-scheme, $$V_Z\to Z,$$ whose base change $V_Z\times_Z E$ is $G$-equivariantly isomorphic to $V_S\times_S E$ as an $E$-scheme. Thus, $V_Z/S$ is a form of $V_S/S$. Since $V\times_S E$ is smooth over $E$, also $V_Z$ is smooth over $B$.

The projection morphism, $$q_{V,E}:V_S\times_S E \to V_Z,$$ is a categorical quotient of $V_S\times_S Z$ by the $G$-action $(\mu_V,\mu_E)$. The natural projection $$\text{pr}_V:V_S\times_S E \to V_S$$ is $G$-equivariant.

Quotient Hypothesis. The $S$-scheme $V_S$ has geometrically irreducible fibers. There exists a categorical quotient $q_V:V_S\to V_S/G$ of the $G$-action. There exists a dense open subscheme $U/G$ of $V_S/G$ whose inverse image $U$ is a $G$-torsor over $U/G$ that is dense in every $S$-fiber of $V_S$.

Nota bene. The hypotheses hold for a generically free linear representation $V_S$ of a finite or geometrically reductive group scheme $G$ (it can also hold for certain representations of non-reductive group schemes).

Now assume that $S$ equals $\text{Spec}(k)$ for an infinite field $k$, assume that $Z$ is affine in a specified class $\mathcal{C}$, let $\Pi\subset Z$ be a closed subscheme that is Artinian, and let $Q\subset V/G$ denote the closed complement of the image of $U$.

Corollary 6. If $V_S/S$ satisfies very weak approximation for class $\mathcal{C}$, then there exists a section $\sigma$ of $V_Z\to Z$ such that the composition $f\circ \sigma$ maps $\Pi$ into the image of $U$.

Proof. This is just Corollary 4 with $R$ set equal to $V/G$. QED

Proposition 7. Let $k$ be an infinite field. Let $G$ be a finite type, affine group $k$-scheme. Let $V_k$ be a smooth, affine $k$-scheme that satisfies very weak approximation for class $\mathcal{C}$ and that satisfies the Quotient Hypothesis. For every affine $k$-scheme $Z$ in class $\mathcal{C}$, for every $G$-torsor $E$ over $Z$, and for every Artinian, closed subscheme $\Pi\subset Z$, there exists an open subscheme $Z'\subset Z$ containing $\Pi$ and a $k$-morphism $s:Z'\to U/G$ such that the $s$-pullback to $Z'$ of the $G$-torsor $U\to U/G$ is isomorphic as a $G$-torsor to the restriction over $Z'$ of the $G$-torsor $E$.

Proof. With respect to the previous corollary, define $Z'$ to be the open complement of $(f\circ \sigma)^{-1}(Q)$, and define $s$ to be the restriction to $Z'$ of $f\circ \sigma$. QED

Corollary 8. For $k$ an infinite field, for every geometrically reductive group $k$-scheme $G$, for every generically free linear representation $V$ of $G$ with free locus $U$, for every affine $k$-scheme $Z$, for every Artinian, closed subscheme $\Pi$ of $Z$, for every $G$-torsor $E$ over $Z$, there exists an open subscheme $Z'$ of $Z$ containing $\Pi$ and there exists a $k$-morphism $s:Z'\to U/G$ such that the $s$-pullback of $U$ is isomorphic as a $G$-torsor to the restriction over $\Pi$ of $E$.

Proof Every generically free linear representation of a geometrically reductive satisfies the Quotient Hypothesis. Thus, the corollary follows from Proposition 8, from Strong Approximation for Affine Space, and from Lemma 3. QED

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    $\begingroup$ Thank you for the amazing answer! Since the names are the same, what is the analogy with your strong/weak approximation and the one I know in the context of p-adic completions of global fields? $\endgroup$ – Scott Foster Oct 4 '18 at 1:14
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    $\begingroup$ @ScottFoster. The analogy is really only for forms of homogeneous spaces. In that case, if $Z$ is the spectrum of the ring of integers of a number field and if $\Pi$ is a varying Artinian closed subscheme, then the definition above should be the same as strong / weak approximation. There are some nice articles of Hassett and Tschinkel about this. I also really like the discussion in Section 2 of Qile Chen -- Yi Zhu, Strong approximation over function fields, arxiv.org/pdf/1510.04647.pdf $\endgroup$ – Jason Starr Oct 4 '18 at 8:20

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