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 $N$.