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Let X be a Brauer-Severi variety over k, i.e. a variety that becomes isomorphic to the projective space after base change to a Galois-field extension of k.

Does the automorphism group Aut(X) act on X transitivelly?

Is there any literature about the action of Aut(X) on X?

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  • $\begingroup$ Please clarify: do you want to know if the automorphism group scheme $G$ of $X$ acts transitively in the sense that the multiplication morphism $(m,\text{pr}_2):G\times_k X \to X\times_k X$ is a surjective morphism of schemes? Or are you asking whether the induced map of sets of $k$-points is surjective, $G(k)\times X(k) \to X(k)\times X(k)$? Of course the answer to both questions is yes, but the second question is a bit silly since $X(k)$ is nonempty if and only if $X$ is isomorphic to $\mathbb{P}^{n-1}_k$. $\endgroup$
    – Jason Starr
    Nov 16, 2012 at 12:13
  • $\begingroup$ I was thinking on something in the sense of transitivity on the closed points of X. So we can consider a finite non split field extension of k, say L, and ask for the transitivity of Aut(X(L))-action on X(L). $\endgroup$
    – Peter
    Nov 16, 2012 at 13:00
  • $\begingroup$ concretely: you have a closed point x in X and a closed point y in X, so is there an automorphism of X such that the automorphism maps x to y? $\endgroup$
    – Peter
    Nov 16, 2012 at 13:10
  • $\begingroup$ Peter: Well, at least the residue fields need to be isomorphic as $k$-algebras, or there is certainly no such automorphism. Maybe a safer, more general question is to ask for a description of the orbits of Aut(X) on the closed points of X. $\endgroup$
    – Pete L. Clark
    Nov 17, 2012 at 0:52

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First of all, I recommend reading Section III.1 of Serre's "Galois Cohomology", particularly III.1.3 on pp. 123-124. In that terminology, the key issue is whether or not the automorphism functor, $\text{Aut}_{X/k}$ in Serre's notation, is representable by a finite type $k$-scheme $G$. Assuming that it is, then the base change of the morphism $(m,\text{pr}_2):G\times_k X\to X\times_k X$ to a field extension $L/k$ is the analogous morphism for the base change $L$-scheme, $X_L = X\times_{\text{Spec} k} \text{Spec} L$. For some (finite) Galois extension $L/k$, there is an isomorphism $X_L \cong \mathbb{P}^{n-1}_L$. For this $L$-scheme, the multiplication morphism $(m,\text{pr}_2)$ is a surjective morphism of finite type $L$-schemes. Since surjectivity can be checked after a faithfully flat base change, this implies that the original morphism $(m,\text{pr}_2)$ over $k$ is surjective. Finally, since this is a morphism of finite type $k$-scheme, every surjective morphism is automatically surjective on closed points: the fiber over every closed point is a nonempty closed subscheme of the domain, which is finite type, hence the closed subscheme contains a closed point of the domain.

Therefore, the main issue is representability of $\text{Aut}_{X/k}$. This is dealt with in many places. The key here is that the dualizing sheaf $\omega_{X/k}$ is anti-ample, i.e., $\omega_{X/k}^\vee$ is ample. The functor $\text{Aut}_{X/k}$ is very close to the functor $\text{Aut}_{X/k,\omega_{X/k}^\vee}$ "parameterizing" a $k$-automorphism $f$ together with an isomorphism between $\omega_{X/k}^\vee$ and $f^*\omega_{X/k}^\vee$. This functor is itself a closed subfunctor of the linear $k$-group of $k$-linear automorphisms of $H^0(X,\omega_{X/k}^\vee)$. There is a discussion of all of this in Section 2.1 of my article with de Jong, "Almost Proper GIT-Stacks and Discriminant Avoidance".

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    $\begingroup$ Regarding representability, couldn't one argue as follows? The base change of $Aut_{X/k}$ to $L$ is $Aut_{X_L/L} \simeq PGL^n_L$, so representability follows from flat descent for affine schemes. $\endgroup$
    – Daniel Bergh
    Nov 19, 2012 at 11:29
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Ok, maybe I am wrong but would this imply that for two closed points x and y in X with isomorphic residue fields there is an automorphism of X ( or an elemet in Aut_{X/k}(k) ) mapping x to y?

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