In this note, we show an explicit concrete generator of the Picard group of any Brauer-Severi variety corresponding to a cyclic algebra in its class inside the Brauer group Br(*k*) of *k*. In particular, for a Brauer-Severi surface *B* and any integer *r* ≥ 1, we obtain a generator for *r* Pic(*B*) from twists of a Fermat type smooth plane curve, see Theorem 4.2. Moreover, we can write equations in ℙ^{9} as follows.

#### Theorem 1.1

*Let B be the Brauer-Severi surface corresponding to a cyclic algebra* (*L*/*k*, *σ*, *a*) *of dimension* 3^{2} *as in Theorem 2.5. A smooth model of B inside* $\begin{array}{}{\mathbb{P}}_{k}^{9}\end{array}$ *is given by the intersection* ∩_{τ∈Gal(L/k)}^{τ}*X where X*/*L is the variety in* $\begin{array}{}{\mathbb{P}}_{L}^{9}\end{array}$ *defined by the set of equations:*

$$\begin{array}{}\begin{array}{c}{a}^{2}({l}_{1}{\omega}_{0}+{l}_{2}{\omega}_{6}+{l}_{3}{\omega}_{9})({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{2}=({l}_{3}{\omega}_{1}+{l}_{1}{\omega}_{5}+{l}_{2}{\omega}_{7}{)}^{3}\\ a({l}_{1}{\omega}_{1}+{l}_{2}{\omega}_{5}+{l}_{3}{\omega}_{7})({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{2}=({l}_{3}{\omega}_{1}+{l}_{1}{\omega}_{5}+{l}_{2}{\omega}_{7}{)}^{2}({l}_{3}{\omega}_{2}+{l}_{1}{\omega}_{3}+{l}_{2}{\omega}_{8})\\ a({l}_{1}{\omega}_{2}+{l}_{2}{\omega}_{3}+{l}_{3}{\omega}_{8})({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{2}=({l}_{3}{\omega}_{1}+{l}_{1}{\omega}_{5}+{l}_{2}{\omega}_{7}{)}^{2}({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9})\\ a({l}_{2}{\omega}_{2}+{l}_{3}{\omega}_{3}+{l}_{1}{\omega}_{8})({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{2}=({l}_{3}{\omega}_{1}+{l}_{1}{\omega}_{5}+{l}_{2}{\omega}_{7})({l}_{3}{\omega}_{2}+{l}_{1}{\omega}_{3}+{l}_{2}{\omega}_{8}{)}^{2}\\ {\omega}_{4}({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{2}=({l}_{3}{\omega}_{1}+{l}_{1}{\omega}_{5}+{l}_{2}{\omega}_{7})({l}_{3}{\omega}_{2}+{l}_{1}{\omega}_{3}+{l}_{2}{\omega}_{8})({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9})\\ a({l}_{2}{\omega}_{0}+{l}_{3}{\omega}_{6}+{l}_{1}{\omega}_{9})({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{2}=({l}_{3}{\omega}_{2}+{l}_{1}{\omega}_{3}+{l}_{2}{\omega}_{8}{)}^{3}\\ ({l}_{2}{\omega}_{1}+{l}_{3}{\omega}_{5}+{l}_{1}{\omega}_{7})({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{2}=({l}_{3}{\omega}_{2}+{l}_{1}{\omega}_{3}+{l}_{2}{\omega}_{8}{)}^{2}({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}),\end{array}\end{array}$$

*and* {*l*_{1}, *l*_{2}, *l*_{3}} *is a non-zero trace (that is, the number* l_{1}+l_{2}+l_{3} ≠ 0) *normal basis of L. Its Picard group* Pic(*B*) *is generated by the intersection of the hyperplane*

$$\begin{array}{}{\omega}_{0}+{\omega}_{6}+{\omega}_{9}=0\end{array}$$

*with B, which is a genus 1 curve over k. More generally, for a positive element r* ∈ Z ≃ Pic(*B*)*, we have a generator of r* Pic(*B*) *given by the intersection of*

$$\begin{array}{}({l}_{1}{\omega}_{0}+{l}_{2}{\omega}_{6}+{l}_{3}{\omega}_{9}{)}^{r}+({l}_{2}{\omega}_{0}+{l}_{3}{\omega}_{6}+{l}_{1}{\omega}_{9}{)}^{r}+({l}_{3}{\omega}_{0}+{l}_{1}{\omega}_{6}+{l}_{2}{\omega}_{9}{)}^{r}=0,\end{array}$$

*with B. It defines a curve of genus* $\begin{array}{}\frac{(3r-1)(3r-2)}{2}\end{array}$ *over k. More precisely, it is k-isomorphic to a twist of the Fermat type curve X*^{3r} + *a*^{r}*Y*^{3r} + *a*^{2r} *Z*^{3r} = 0.

Several people worked on finding (or trying to find) equations for Brauer-Severi varieties: using ideas of Châtelet (cf. [7, 11]), the Grothendieck descent (cf.[6]), Grassmanians (cf. [5]) and special embeddings in the projective space (cf. [§5.2]GS, [9]). All these constructions lack in how to explicitly construct subvarieties of codimension 1 inside them, that is, elements of their Picard group. Accordingly, we are motivated to find curves’ equations for generators of the subgroups *r* Pic(*B*). This is what we do in Theorem 1.1 when *B* is a Brauer-Severi surface, and in Theorem 6.2 for higher dimensional Brauer-Severi varieties (at least for the ones associated to cyclic algebras).

The key idea of this paper is inspired by [1, 10] and the theory of twists, where any fixed twist of a smooth plane curve is embedded into a certain Brauer-Severi surface that becomes trivial (*k*-isomorphic to ℙ^{2}) if and only if that twist has a smooth plane model over *k*.

In general, we attach to a cocycle *ξ* ∈ H^{1} (*k*, PGL_{n+1} (*k*)) coming from a cyclic algebra, a Brauer-Severi variety of dimension *n* together with a codimension 1 subvariety living inside it (in the case *n* = 2, this subvariety is a twist of a smooth plane curve).

Here we consider any Brauer-Severi variety *B* associated to some cyclic algebra and then we determine a family of smooth hypersurfaces such that some of their twists are embedded into *B*. We start by choosing the automorphism group properly (a cyclic group of automorphisms of specified shapes, related to the cyclic algebra we already have). This in turns allows us to conclude that certain twists produce generators for the subgroups *r* Pic(*B*).

We obtain explicit equations for the Brauer-Severi varieties *B* and for the aforementioned generators in Section 5, 6 and 7. The difference between the approaches in the different sections is the map we use in Galois cohomology transporting the cocycle *ξ* to a trivial cocycle in another Galois cohomology set. In Section 5 and 6, we use the Veronese embedding while in Section 7 we use the canonical embedding corresponding to a certain smooth plane curve *C* such that we can see *ξ* ∈ H^{1} (*k*, Aut(*C*)).

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