The Picard Group of Brauer-Severi Varieties

In this note we provide explicit generators of the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular, for all Brauer-Severi surfaces. To produce these generators we use the Twisting Theory for smooth plane curves.


Introduction
Let B/k be a Brauer-Severi variety over a perfect field k, that is, a projective variety of dimension n isomorphic over k to P n k .Its group the Picard Pic(B) it is known to be isomorphic to Z.As far as we know, the first explicit equations defining a non-trivial Brauer-Severi surface in the literature are in [1].After this, an algorithm to compute these equations for any Brauer-Severi variety is given in [7].In the appendix, we explain an alternative way to compute them for the case of dimension 2 and using twists of smooth plane curves.
In this note, we show an explicit and 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.More precisely, for Brauer-Severi surfaces we obtain the folowwing result.

Brauer-Severi varieties
Definition 2.1.Let V be a smooth quasi-projective variety over k.A variety E. Badr and F. Bars are supported by MTM2016-75980-P.
Theorem 2.2.[9, Ch.III, §1.3]Following the above notations, for any Galois extension K/k, there exists a bijection where Aut K (.) denotes the group of K-automorphisms of the object over K.
For K = k, the right hand side will be denoted by H 1 (k, Aut k (V )) or simply H 1 (k, Aut(V )).

trivial if and only if a is the norm of an element of L
This Theorem is a conclusion from the fact that H 1 (k, PGL n (k)) is in correspondence with the set Az n k of central simple algebras of dimension n 2 over k, modulo k-isomorphisms [10, chap.X.5], the fact that Az 3 k contains only cyclic algebras [12] and the description of cyclic central simple algebras given in [11,Example 5.5].For the last statement see [3, §2.1].

Smooth plane curves
Fix an algebraic closure k of a perfect field k.By a smooth plane curve C over k of degree d ≥ 3, we mean a curve C/k, which is k-isomorphic to the zero-locus of a homogenous polynomial equation k be a morphism given by (the unique) g 2 d -linear system over k, then there exists a Brauer-Severi variety B (of dimension two) defined over k, together with a k-morphism f : k is equal to Υ.In [1] we constructed twists of smooth plane curves over k not having smooth plane model over k.These twists happened to be contained in Brauer-Severi surfaces as in Theorem 3.1.
Theorem 3.2.[Theorem 3.1 in [1]] Given a smooth plane curve over k: C/k ⊆ P 2 with degree d ≥ 4, there exists a natural map ) is the set of twists of C admitting a smooth plane model over k, where [P 2 k ] is the trivial class associated to the trivial Brauer-Severi surface of the projective plane over k.Remark 3.3 (Remark 3.2 in [1]).We can reinterpret the map Σ in Theorem 3.2 as the map that sends a twist C ′ to the Brauer-Severi variety B in Theorem 3.1.
These results suggest the opposite question, instead of given the curve C and the twist C ′ and finding the Brauer-Severi surface B, fixing the Brauer-Severi surface B and trying to find the right curve C and the right twist C ′ to find the k-morphism f : The main idea is looking for smooth plane curves C of degree a multiple of 3, otherwise all their twists are smooth plane curves over k, see Theorem 2.6 in [1], and having an automorphism of the form [aZ : X : Y ] to define the twist C ′ given by the cocycle that sends a generator σ of the degree 3 cyclic extension L/k defining B to the automorphism [aZ : The map δ sends 1 to the Brauer class corresponding to B. Proof.The twist C ′ of C d ′ a given by the inflation map of the cocycle as in Theorem 2.5 lives inside B for any integer d ′ ≥ 2 by using Theorem 3.2 with Remark 3.3.For d ′ = 1, set Aut L (C 1 a ) for the subgroup of automorphisms of C 1 a acting linearly on the variables X, Y, Z. Therefore, the inclusions Aut L (C 1 a ) ≤ PGL 3 (k) and Aut L (C 1 a ) ≤ Aut(C) give us the two natural maps L : respectively.Second, compose with the 3-Vernoese embedding Ver 3 : P 2 k → P 9 k , to obtain a model of C inside the trivial Brauer-Severi surface Ver 3 (P 2 k ).Because the image of any 1-cocycle by the map Ṽer 3 : H 1 (k, PGL 3 (k)) → H 1 (k, PGL 10 (k)), is equivalent to a 1-cocycle with values in GL 10 (k) [7], and a over k associated to ξ by Theorem 2.5.On the other hand, by Wedderburn [12] and Theorem 4.1, the map δ sends 1 to the Brauer class [B] of B inside the 3-torsion Br(k) [3] of the Brauer group Br(k) of the field k.Hence [B] has exact order 3, being non-trivial, and so Pic(B) inside Pic(B k has degree 3d ′ , hence it corresponds to the ideal (3d ′ ) ⊂ Z via the degree map.Consequently, the image of C ′ in Pic(B) is a generator of d ′ Pic(B).
On the other hand, the twist φ : given by the previous cocycle is embedded in B: we have the k-morphism f : C ′ → B given by φ −1 Υφ.Composing with Ver 3 we get the equations of C ′ inside P 9 in the statement of Theorem 1.1.Finally, the claim about the order of the curves C ′ in Pic(B) follows by Theorem 4.2.

Generalizations on Picard group elements for cyclic Brauer-Severi varieties
Let L/k be a Galois cyclic extension of degree n + 1, with Gal(L/k) = σ .Fix a character which is equivalent to fix σ ′ a generator of Gal(L/k) such that χ(σ ′ ) = 1.Given a ∈ k * , we consider a k-algebra (L/k, χ, a) as follows: As an additive group, (L/k, χ, a) is an n + 1-dimensional vector space over L with basis 1, e, . . ., e n : (L/k, χ, a) := ⊕ n i=0 Le i with 1 = e 0 .Multiplication is given by the relations: e .λ = σ ′ (λ) .e for λ ∈ L, and e n+1 = a.The algebra (L/k, χ, a) is called the cyclic algebra associated to the character χ and the element a ∈ k, and is trivial if and only if a is a norm of certain element of L. Its class in H 1 (k, PGL n+1 (k)) corresponds to the inflation of the cocycle in H 1 (Gal(L/k), PGL n+1 (L)) given by Theorem 6.2.Let B be a Brauer-Severi variety over k, associated to a cyclic algebra (L/k, χ, a) of dimension (n + 1) 2 and exact order n + 1 in Br(k).For any integer d ′ ≥ 1, there is a twist X ′ over k of X d ′ ,n a , living inside B and defining a generator of d ′ Pic(B). .Use the n-Veronese embedding Ver n to obtain a model of X d ′ ,n a inside the trivial Brauer-Severi variety Ver n (P n k ).Because the image of a 1cocyle by the map Ṽer n : H 1 (k, PGL n+1 (k)) → H 1 (k, PGL m+1 (k)) is equivalent to a 1-cocycle with coefficients in GL m+1 (k) by [7] and Following the notation of [7, Lemma 3.1], we write V n : P n → P m : (X 0 : ... : X n ) → (ω 0 : ... : ω m ), where the ω k are equal to the products ω By using [6, Section 3], we find that a matrix φ realizing the cocycle ξ, that is, We plug φ into the equation of X d ′ ,n a and the result follows.

Appendix: Another approach to construct Brauer-Severi surfaces
The third author shows an algorithm for constructing equations of Brauer-Severi varieties in [7].Here we show an alternative way for constructing equations of Brauer-Severi surfaces (n = 2) by using the Twisting Theory of plane curves.

Let Ver
), satisfies that the image of any 1-cocycle is equivalent to a 1-cocycle with values in the lineal group GL ( 2n−1 n−1 ) (k) and is well-know that H 1 (k, GL ( 2n−1 n−1 ) (k)), is trivial by Hilbert 90 Theorem.This fact leads to an algorithm to compute equations for any Brauer-Severi varieties.Here we use the idea coming from the construction in [1] of the equations for a non-trivial Brauer-Severi surface., where ι comes from the (unique) g 2 d -linear system, all are defined over k.In particular, fixing a non-singular plane model F C (X, Y, Z) = 0 in P 2 k of C, one may directly compute its canonical embedding into P

Theorem 4 . 2 .
Let B be a non-trivial Brauer-Severi surface over k, associated to a cyclic algebra (L/k, χ, a) of dimension 9 by Theorem 2.5.For any integer d ′ ≥ 1, there is a twist C ′ over k of the smooth plane curve C d ′ ,a , living inside B and also defines a generator of d ′ Pic(B).

Proof.
Set m = 2n+1 n and consider the Veronese embedding Ver n : P n k ֒→ P m−1 k

a
over k associated to ξ : σ → A σ .On the other hand, by Theorem 4.1, the map δ sends 1 to the Brauer class [B] of B inside the (n + 1)-torsion Br(k)[n + 1] of the Brauer group Br(k) of the field k.Hence [B] has exact order n + 1, being non-trivial, and so Pic(B) inside Pic(B ⊗ k k = P n k ) deg ∼ = Z is isomorphic to (n + 1)Z.Moreover, X ′ × k k ⊆ P n k has degree (n + 1)d ′ , hence it corresponds to the ideal ((n + 1)d ′ ) ⊂ Z via the degree map.Consequently, the image of X ′ in Pic(B) is a generator of d ′ Pic(B).

1 k
be the n-Veronese embedding.It has been observed by the third author in[7] that the induced map Ṽer
Le ⊕ Le 2 .Multiplication is given by the relations: e .λ = σ ′ (λ) .e for λ ∈ L, and e 3 = a.The algebra (L/k, χ, a) is called the cyclic algebra associated to the character χ and the element a ∈ k.
Theorem 2.5.Any non-trivial Brauer-Severi surface B over k corresponds, modulo k-isomorphism, to a cyclic algebra (L/k, χ, a) of dimension 9, for some Galois cubic extension L/k and a ∈ k * , which is not a norm of an element defines a smooth plane curve C d ′ [2,ver k of degree 3d ′ , such that [aZ : X : Y ] is an automorphism.4.The Picard groupTheorem 4.1.(Lichtenbaum,see[2,Theorem 5.4.10])Let B be a Brauer-Severi variety over k.Then, there is an exact sequence