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Let $X$ be a projective variety over a perfect field $k$. Recall that a twist of $X$ is a variety $Y$ over $k$ such that $$X_{\bar k} \cong Y_{\bar k}.$$ The twists of $X$ are classified by the Galois cohomology set $\mathrm{H}^1(k,\mathrm{Aut} X_{\bar k}).$

My question concerns what happens when one considers projective automorphisms instead. Namely, fix an embedding $X \subset \mathbb{P}^n$ and let $\mathrm{PAut} X$ denote the collection of automorphisms of $X$ which are induced by an automorphism of the ambient projective space.

What does $\mathrm{H}^1(k,\mathrm{PAut} X_{\bar k})$ classify?

A first naive guess would be that this classifies varieties $Y \subset \mathbb{P}^n$ which become projectively isomorphic to $X$ over $\bar k$; however this is clearly seen to be false on taking $X = \mathbb{P}^n$.

It might help to put this problem into a more general framework. Namely, let $(X,L)$ be a projective variety equipped with a line bundle $L$. Denote by $\mathrm{Aut}(X,L)$ those automorphisms of $X$ which preserve the isomorphism class of $L$.

What does $\mathrm{H}^1(k,\mathrm{Aut}(X_{\bar k},L_{\bar k}))$ classify?

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  • $\begingroup$ What precisely do you mean by "preserve the isomorphism class"? For instance, it seems to me that $\text{Aut}(\mathbb{P}^n_k,\mathcal{O}(1))$ is $\textbf{PGL}_{n+1}$. There is a slightly different notion that recovers $\textbf{SL}_{n+1}$. Some of this is discussed in my paper with de Jong, "Discriminant avoidance ..." $\endgroup$ – Jason Starr Mar 20 '15 at 12:31
  • $\begingroup$ I mean those $\sigma \in \mathrm{Aut} X$ such that $\sigma^*L \cong L$. Certainly I want $\mathrm{Aut}(\mathbb{P}^n, \mathcal{O}(1) )$ to be $\mathrm{PGL}_{n+1}$. $\endgroup$ – Daniel Loughran Mar 20 '15 at 13:17
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    $\begingroup$ It seems to me that the "obvious guess" to your first question would be isomorphism classes of pairs $(Y,Z)$ so that $Y$ becomes isomorphic to $X$ and $Z$ becomes isomorphic to $\mathbb{P}^n$ over $\bar{k}$. For general line bunbdles this seems trickier... $\endgroup$ – ulrich Mar 23 '15 at 9:13
  • $\begingroup$ I should have assumed that $X$ is non-degenerate (so there is at most one automorphism of $\mathbb{P}^n$ inducing a given automorphism of $X$). $\endgroup$ – ulrich Mar 24 '15 at 5:03
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Thanks to the hint from Ulrich, I think I am now able to answer the question.

The set $\mathrm{H}^1(k, \mathrm{Aut}(X_{\bar k},L_{\bar k}))$ classifes the following objects:

$k$-Isomorphism classes of pairs $(Y,E)$, where $Y$ is a projective variety over $k$ and $E \in \mathrm{Pic}_{Y/k}(k)$, which become isomorphic to $(X,L)$ after a finite separable field extension.

Here $\mathrm{Pic}_{Y/k}$ denotes the Picard scheme of $Y$. Note that we have $$\mathrm{Pic}_{Y/k}(k) = \mathrm{Pic}(\bar Y)^{\mathrm{Gal}(\bar k/ k)}.$$

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