Let $k$ be a field of characteristic $0$ und let $X$ be a del Pezzo surface over $k$. Note that $X$ may not have points.

Let us consider $N:=\ker(\mathrm{Br}(k) \rightarrow \mathrm{Br}(k(X))$.

Question 1: Under which circumstances depending on $k$ or $X$, is $N$ generated by one element?

From the Hochschild-Serre spectral sequence, we obtain the infamous sequence

$0 \rightarrow \mathrm{Pic(X)} \rightarrow \mathrm{Pic}(\overline{X})^{\mathrm{Gal}(\overline{k}/k)} \rightarrow \mathrm{Br}(k)$.

The last map assigns a Brauer class to each element in $\mathrm{Pic}(\overline{X})^{\mathrm{Gal}(\overline{k}/k)}$ , which I will call a Tits algebra. Clearly, depending on the rank of $\mathrm{Pic}$, there may be several Tits algebras. It is possible that $X$ has non trivial Tits algebras. Also they may be all isomorphic.

We denote the Severi-Brauer variety of a central, simple algebra $A$ by $\mathrm{SB}(A)$.

Question 2: Assume that $X$ has none trivial Tits algebras. How do they change over $k(X)$ ?

Question 3: Assume $X$ has only one Tits algebra $A$, which is not trival and let us assume that $N$ is generated by one element. Does $N$ become trivial over $k(\mathrm{SB}(A))$? What is the general relation between $A$ and the generator of $N$.

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    $\begingroup$ Please consider using \operatorname, \mathrm, \ker, etc. This looks pretty bad. $\endgroup$ – RP_ Aug 3 at 16:13
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    $\begingroup$ Except for $\ker$ , which I simply screwed up, I dont think there is a convention for this, as I know several examples of writing this stuff both ways. But since i am writing down my thesis at this moment, I thank you. I will grind through the papers of the reviewers to make sure I match their standards. $\endgroup$ – nxir Aug 3 at 20:18
  • $\begingroup$ Well, many people are doing it wrong. The spacing alone indicates that when you write something like $ker$, it is meant to be read as $k$ times $e$ times $r$. If you want to convince yourself, I think the convention is upright letters for Gal, Br, Hom, etc. in the great majority of books. Also if you sample some publications by authors who give evidence of paying attention to typographical matters, you will see that they are all doing this. Anyway, thanks for fixing it, changed my vote from -1 to +1. $\endgroup$ – RP_ Aug 3 at 23:01

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