The Picard group of Brauer-Severi varieties

Eslam Badr; Francesc Bars; Elisa Lorenzo García

doi:10.1515/math-2018-0101

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The Picard group of Brauer-Severi varieties

Eslam Badr , Francesc Bars and Elisa Lorenzo García

Published/Copyright: October 29, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

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

Keywords: Brauer-Severi varieties; Picard groups; Cyclic Algebras; Theory of Twists; Automorphisms

MSC 2010: 11G35; 14J26; 11D41; 14J50; 14J70; 11R34

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 Pkn. The Picard Pic(B) 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 [9]. In the appendix, we explain an alternative way to compute them for the case of dimension 2 by using twists of smooth plane curves.

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 ℙ⁹ as follows.

Theorem 1.1

Let B be the Brauer-Severi surface corresponding to a cyclic algebra (L/k, σ, a) of dimension 3²as in Theorem 2.5. A smooth model of B insidePk9is given by the intersection ∩_{τ∈Gal(L/k)}^τX where X/L is the variety inPL9defined by the set of equations:

a2(l1ω0+l2ω6+l3ω9)(l3ω0+l1ω6+l2ω9)2=(l3ω1+l1ω5+l2ω7)3a(l1ω1+l2ω5+l3ω7)(l3ω0+l1ω6+l2ω9)2=(l3ω1+l1ω5+l2ω7)2(l3ω2+l1ω3+l2ω8)a(l1ω2+l2ω3+l3ω8)(l3ω0+l1ω6+l2ω9)2=(l3ω1+l1ω5+l2ω7)2(l3ω0+l1ω6+l2ω9)a(l2ω2+l3ω3+l1ω8)(l3ω0+l1ω6+l2ω9)2=(l3ω1+l1ω5+l2ω7)(l3ω2+l1ω3+l2ω8)2ω4(l3ω0+l1ω6+l2ω9)2=(l3ω1+l1ω5+l2ω7)(l3ω2+l1ω3+l2ω8)(l3ω0+l1ω6+l2ω9)a(l2ω0+l3ω6+l1ω9)(l3ω0+l1ω6+l2ω9)2=(l3ω2+l1ω3+l2ω8)3(l2ω1+l3ω5+l1ω7)(l3ω0+l1ω6+l2ω9)2=(l3ω2+l1ω3+l2ω8)2(l3ω0+l1ω6+l2ω9),

and {l₁, l₂, l₃} is a non-zero trace (that is, the number l₁+l₂+l₃ ≠ 0) normal basis of L. Its Picard group Pic(B) is generated by the intersection of the hyperplane

ω0+ω6+ω9=0

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

(l1ω0+l2ω6+l3ω9)r+(l2ω0+l3ω6+l1ω9)r+(l3ω0+l1ω6+l2ω9)r=0,

with B. It defines a curve of genus(3r−1)(3r−2)2over k. More precisely, it is k-isomorphic to a twist of the Fermat type curve X^3r + a^rY^3r + a^2rZ^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 ℙ²) if and only if that twist has a smooth plane model over k.

In general, we attach to a cocycle ξ ∈ H¹ (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 ∈ H¹ (k, Aut(C)).

2 Brauer-Severi varieties

Definition 2.1

Let V be a smooth quasi-projective variety over k. A variety V′ defined over k is called a twist of V over k if there is ak-isomorphism V′×kk¯⟶ΦV¯:=V×kk¯.The set of all twists of V modulo k-isomorphisms is denoted by Twist_k (V), whereas the set of all twists V′ of V over k, such that V′ × _kK is K-isomorphic to V × _kK is denoted by Twist(V, K/k).

Theorem 2.2 ([12, Chp. III, §1.3])

Following the above notations, for any Galois extension K/k, there exists a bijection

θ:Twist⁡(V,K/k)→H1⁡(Gal⁡(K/k),AutK⁡(V×kK))V′×kK≅ΦV×kK↦ξ(τ):=Φ∘τΦ−1

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 H¹ (k, Aut_k(V)) or simply H¹ (k, Aut(V)).

Definition 2.3

A Brauer-Severi variety B over k of dimension n is a twist ofPkn. The set of all isomorphism classes of Brauer-Severi varieties of dimension n over k is denoted byBSnk.

Corollary 2.4. ([6, Corollary 4.7])

The setBSnkis in bijection withTwistk⁡(Pkn)=H1⁡(k,PGLn+1⁡(k¯)).

2.1 Brauer-Severi surfaces

Let L/k be a Galois cyclic cubic extension and let σ be a fixed generator of the Galois group Gal(L/k). Given a ∈ k∗, we may consider a k-algebra (L/k, σ, a) as follows: As an additive group, (L/k, σ, a) is a 3-dimensional vector space over L with basis 1, e, e²: (L/k, σ, a):=L ⊕ Le ⊕ Le². Multiplication is given by the relations: e.λ = σ (λ).e for λ ∈ L, and e³ = a. The algebra (L/k, σ, a) is called the cyclic algebra associated to σ 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 of dimension 9 of the form (L/k, σ, a), for some Galois cubic extension L/k and a ∈ k∗ which is not a norm of an element of L. If Gal(L/k) = 〈σ〉, then the image of B in H¹ (k, PGL₃ (k)) is given by

ξ(σ)=00a100010.

Moreover, the Brauer-Severi surface attached to (L/k, σ, a) ∈ H¹ (k, PGL₃ (k)) is trivial if and only if a is the norm of an element of L.

The above theorem (Theorem 2.5) can be concluded from the fact that H¹ (k, PGL_n (k)) is in correspondence with the set Aznk of central simple algebras of dimension n² over k, modulo k-isomorphisms [13, Chp. X.5], the fact that Az3k contains only cyclic algebras [15], and the description of cyclic central simple algebras given in [2, Construction 2.5.1 and Proposition 2.5.2] (see also [14, Example 5.5]). For the last statement, we refer to [3, §2.1].

3 Smooth plane curves

Fix an algebraic closure k of a perfect field k. By a smooth plane curve Cover k of degree d ≥ 3, we mean a curve C/k, which is k-isomorphic to the zero-locus in Pk¯2 of a homogenous polynomial equation F_C (X, Y, Z) = 0 of degree d with coefficients in k, that has no singularities. In that case, the geometric genus of C equals g=12(d−1)(d−2). Assuming that d ≥ 4, the base extension C × _kk admits a unique gd2-linear system up to conjugation in Aut⁡(Pk¯2)=PGL3⁡(k¯). It induces a unique embedding Υk¯:C¯→Pk¯2, up to PGL₃ (k)-conjugation, giving a Gal(k/k)-equivariant map Aut(C¯¯) ↪ PGL₃ (k).

Theorem 3.1. (Roé-Xarles, [10])

Let C be a curve over k such thatC = C × _kkis a smooth plane curve overkof degree d ≥ 4. LetΥk¯:C¯ ↪ Pk¯2be a morphism given by (the unique)gd2-linear system overk, then there exists a Brauer-Severi surface B defined over k, together with a k-morphism f:C ↪ B such that f×kk¯:C¯→Pk¯2is equal to Υ_k.

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 non-trivial Brauer-Severi surfaces as in Theorem 3.1.

Theorem 3.2. ([1, Theorem 3.1])

Given a smooth plane curveC⊆Pk¯2over k of degree d ≥ 4, there exists a natural map

Σ:H1⁡(k,Aut⁡(C))→H1⁡(k,PGL3⁡(k¯)),

moreoverΣ−1([Pk2])is the set of twists for C that admit a smooth plane model over k. Here[Pk2]denotes the trivial class associated to the trivial Brauer-Severi surface of the projective plane over k.

Remark 3.3 ([1, Remark 3.2])

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 giving the curve C and the twist C′, and then finding the Brauer-Severi surface B, fix the Brauer-Severi surface B and try to find the right curve C and the right twist C′ in order to establish the k-morphism f:C′ ↪ B.

The main idea is to look for smooth plane curves C of degree divisible by 3, otherwise all their twists are smooth plane curves over k by [Theorem 2.6], and that have an automorphism of the form [aZ : X : Y]. Next, to consider the twist C′ given by the cocycle defining B, which sends a certain generator σ of the degree 3 cyclic extension L/k to the automorphism [aZ : X : Y].

Lemma 3.4

For any a ∈ k∗ and r ∈ Z_{≥ 1}, the equation X^3r + a^rY^3r + a^2rZ^3r = 0, defines a smooth plane curveCarover k of degree 3r, such that [aZ : X : Y] is an automorphism.

4 The Picard group

Theorem 4.1. (Lichtenbaum, see [2, Theorem 5.4.10])

Let B be a Brauer-Severi variety over k. Then, there is an exact sequence

0⟶Pic⁡(B)⟶Pic⁡(B×kk¯)≅degZ⟶δBr(k).

The map δ sends 1 to the Brauer class corresponding to B.

Theorem 4.2

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

Proof

We conclude by the virtue of Theorem 3.2 and Remark 3.3 that the twist C′ of Car given by the inflation map of the cocycle

ξ(σ)=00a100010∈H1(Gal⁡(L/k),Aut⁡(Car¯))

as in Theorem 2.5 lives inside B for any integer r ≥ 2. For r = 1, set Autinv⁡(Ca1) for the subgroup of automorphisms of Ca1, leaving invariant the equation X³ + aY³ + a²Z³ = 0. Therefore, the inclusions Autinv⁡(Ca1)≤PGL3⁡(k¯) and Autinv⁡(Ca1)≤Aut⁡(C¯) give us the two natural maps inv:H1⁡(k,Autinv⁡(Ca1))→H1⁡(k,Aut⁡(Ca1¯)), and Σ:H1⁡(k,Autinv⁡(Ca1))→H1⁡(k,PGL3⁡(k¯)) respectively. Second, compose with the 3-Veronese embedding Ver3:Pk2→Pk9 to obtain a model of C inside the trivial Brauer-Severi surface Ver3⁡(Pk2). Because the image of any 1-cocycle by the map Ver~3:H1⁡(k,PGL3⁡(k¯))→H1⁡(k,PGL10⁡(k¯)), is equivalent to a 1-cocycle with values in GL₁₀ (k), see [9], and since H¹ (k, GL₁₀ (k)) = 1, then Ver~3([B]) is given in H¹ (k, PGL₁₀ (k)) by τ ∈ Gal(k/k)↦M ∘ ^τM⁻¹, for some M ∈ GL₁₀ (k). Consequently, (M∘Ver3)(Pk2) is a model of B in Pk9, containing (M∘Ver3)(Ca1) inside, which is a twist of Ca1 over k associated to ξ by Theorem 2.5.

On the other hand, due to the results of Wedderburn in [15] 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×kk¯=Pk¯2)≅deg(Z) is isomorphic to 3ℤ. Moreover, C′×kk¯⊆Pk¯2 has degree 3r , hence it corresponds to the ideal 〈3r 〉⊂ℤ via the degree map. Consequently, the image of C′ in Pic(B) is a generator of r Pic(B).

5 The proof of Theorem 1.1

Let B be the Brauer-Severi surface corresponding to (L/k, σ, a) as in Theorem 2.5. Then, there is an isomorphism ϕ¯:B×kL→PL2 defined over L such that

ξ(σ)=00a100010=ϕ¯.σϕ¯−1.

The results in [9] would be applied to get the equations in the statement of Theorem 1.1 for B inside ℙ⁹. We recall that the equations are obtained by twisting the image of ℙ² into ℙ⁹ by the Veronese embedding Ver₃:ℙ² → ℙ⁹. Indeed, we can compute following [9]

Ver3⁡(ϕ¯)=a2l100000a2l200a2l30al1000al20al30000al1al20000al3000al2al30000al1000001000000l3000l10l200al200000al300al10l2000l30l10000l3l10000l20l300000l100l2:B×kL→PL2⊆P9,

where L = k (l₁, l₂, l₃) with σ (l₁) = l₂ and σ (l₂) = l₃.

On the other hand, the twist ϕ:C′→Car given by the previous cocycle is embedded in B: we have the k-morphism f : C′ → B arisen from Theorem 3.1 through the L-morphism ϕ^—1 ∘ Υ_L ∘ ϕ × _kL. Composing with Ver₃ we get the equations of C′ inside ℙ⁹in the statement of Theorem 1.1.

Finally, the claim about the order of the curves C′ in Pic(B) follows from Theorem 4.2.

6 Generalizations on Picard group elements for cyclic Brauer-Severi varieties

Let L/k be a Galois cyclic extension of degree n + 1 and fix a generator σ for Gal(L/k). Given a ∈ k∗, one considers a k-algebra (L/k, σ, a) as follows: As an additive group, (L/k, σ, a) is an (n + 1)-dimensional vector space over L with basis 1,e,…,en:(L/k,σ,a):=⊕i=0nLei with 1 = e⁰. Multiplication is given by the relations: e⋅λ = σ (λ) ⋅ e for λ ∈ L, and e^{n + 1} = a. The algebra (L/k, σ, a) is called the cyclic algebra associated to σ and the element a ∈ k. It is trivial if and only if a is a norm of certain element of L. Its class in H¹ (k, PGL^{n + 1} (k)) corresponds to the inflation of the cocycle in H¹(Gal(L/k), PGL^{n + 1}(L)) given by

ξ(σ)=00……0a10⋱⋱00010⋱⋮⋮⋮⋱⋱⋱⋮00……10.

For more details, one may read [2, Construction 2.5.1 and Proposition 2.5.2].

Lemma 6.1

For any a ∈ k∗ and r ∈ ℤ_≥1, the equation

∑i=0nairXi(n+1)r=0

defines a non-singular k-projective modelXar,nof degree (n + 1)r of a smooth projective variety insidePkn, such that A_a:= [aX_n:X₀:…:X_n−1] is leaving invariant Xar,n.

Theorem 6.2

Let B be a Brauer-Severi variety over k, associated to a cyclic algebra (L/k, σ, a) of dimension (n + 1)² and exact order n + 1 in Br(k). For any integer r ≥ 1, there is a twist X′ over k of Xar,n, living inside B and defining a generator of r Pic(B).

Proof

Set m=2n+1n−1 and denote by Autinv⁡(Xar,n) the subgroup of automorphisms of Xar,n, leaving invariant its defining equation. Therefore, the inclusions Autinv⁡(Xar,n)≤PGLn+1⁡(k¯) and Autinv⁡(Xar,n)≤Aut⁡(Xar,n×kk¯) give us the two natural maps inv:H1⁡(k,Autinv⁡(Xar,n))→H1⁡(k,Aut⁡(Xar,n×kk¯)), and Σ:H1⁡(k,Autinv⁡(Xar,n×kk¯))→H1⁡(k,PGLn+1⁡(k¯)) respectively. Compose with the n-Veronese embedding Vern:Pkn→Pkm, to obtain a model of Xar,n inside the trivial Brauer-Severi variety Vern⁡(Pkn). Because the image of a 1-cocyle under the map Ver~n:H1⁡(k,PGLn+1⁡(k¯))→H1⁡(k,PGLm+1⁡(k¯)) is equivalent to a 1-cocycle with coefficients in GL_{m + 1} (k) by [9] and since H¹ (k, GL_{m + 1} (k)) = 1, then Ver~3([B]) is given in H¹ (k, PGL_{m + 1} (k)) by τ ∈ Gal(k/k) ↦ M ∘ ^τM⁻¹, for some M ∈ GL_{m + 1} (k). Consequently, (M∘Ver3)(Pk2) is a model of B in Pk9, that contains (M∘Ver3)(Ca1) inside, which is a twist of Xar,n over k associated to ξ : σ ↦ A_a.

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×kk¯=Pk¯2)≅deg(Z) is isomorphic to (n + 1)ℤ. Moreover, X′×kk¯⊆Pk¯2 has degree (n + 1)r, hence it corresponds to the ideal 〈(n + 1)r 〉 ⊂ ℤ via the degree map. Consequently, the image of X′ in Pic(B) is a generator of r Pic(B).

Following the notation of [9, Lemma 3.1], we write Vn:Pn→Pm:(X0:...:Xn)↦(ω0:...:ωm), where the ω_k are equal to the products ωX0α0…Xnαn=∏iXiαi with ∑_iα_i = n + 1 in alphabetical order. The automorphism of Xar,n as an automorphism of Vern⁡(Xar,n) sends ωXin↦ωXi+1n and ωXnn↦aωX1n.

Corollary 6.3

With the notation above,

Vern⁡(X′):∑i=0n∑jli+jωXjnr=0⊆B

Proof

By using [8, §3], we find that a matrix ϕ realizing the cocycle ξ, that is, ξ = ϕ ∘ ^σϕ⁻¹, sends ϕ(ωXin)=ai∑jli+jωXjn and ϕ(ωXnn)=∑jlj−1ωXjn. We plug ϕ into the equation of Xar,n and the result follows. □

7 Comparison for constructing Brauer-Severi surfaces via canonical embedding of smooth plane curves

The third author shows an algorithm for constructing equations of Brauer-Severi varieties in [9]. Here we show an alternative way for constructing equations of Brauer-Severi surfaces (n = 2) by using the theory of twists of plane curves.

Let Vern:Pkn↪Pk2n+1n−1 be the n-Veronese embedding. It has been observed by the third author in [9] that the induced map

Ver~n:H1⁡(k,PGLn+1⁡(k¯))→H1⁡(k,PGL2n+1n⁡(k¯)),

satisfies that the image of any 1-cocycle is equivalent to a 1-cocycle with values in the lineal group GL2n+1n⁡(k¯) and it is well-known that H1⁡(k,GL2n+1n⁡(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.

Lemma 7.1

Let C be a smooth plane curve over k of genusg=12(d−1)(d−2)≥3. The canonical embedding of C is isomorphic to the compositionΨ:C⟶ιPk2⟶Verd−3Pkg−1,where ι comes from the (unique) gd2-linear system, all are defined over k. In particular, fixing a non-singular plane model F_C (X, Y, ℤ) = 0 inPk2 of C, one may directly compute its canonical embedding intoPkg−1by applying the morphism Ver_d−3.

Proof

It is fairly well-known that the sheaves Ω¹(C) and O(d − 3)|_C are isomorphic (cf. R. Hartshorne [4, Example 8.20.3]). Hence, H⁰(ℙ², O(d − 3)) ⟶ H₀ (C, Ω₁) is an isomorphism, and the statement follows. □

Both maps, ι and Ver_{d − 3}, are Gal(k/k)-equivariant. Therefore, the natural maps

Aut⁡(C¯)↪Aut⁡(Pk¯2)=PGL3⁡(k¯)→Aut⁡(Pk¯g−1)=PGLg⁡(k¯)

are morphisms of Gal(k/k)-groups.

Proposition 7.2

Given a non-trivial Brauer-Severi surface B over k, associated to a cyclic algebra (L/k, σ, a) of dimension 9, a k-model ofPk92r(r−1)with r ≥ 2 is algorithmically computable by considering any smooth plane curve C over k of degree 3r such that [aZ : X : Y] is an automorphism. In particular, for r = 2 we get a model^[1] in ℙ⁹.

Proof

For non-hyperelliptic curves, see a description in [8], the canonical model gives a natural Gal(k/k)-inclusion Aut(C¯¯) ↪ PGL_g (k), but we can go further, the action gives a Gal(k/k)-inclusion Aut(C¯¯) ↪ GL_g (k). In this way, the natural map H¹ (k, Aut(C¯¯)) → H¹ (k, PGL_g (k)), satisfies that the image of any 1-cocycle is equivalent to a 1-cocycle with values in GL_g (k), and recall that H¹ (k, GL_g (k)) is trivial by applying Hilbert’s Theorem 90. This allows us to compute equations for twists via change of variables in GL_g (k) of the canonical model for C. Now, by Lemma 7.1 and the proof of Theorem 4.2, one could construct a smooth model for the Brauer-Severi surface in Pk9 by taking the V_{d − 3} embedding with d = 6 and C as in the statement. □

Acknowledgement

We thank the anonymous referee for helping us improving the clarity of the exposition and his/her useful comments and suggestions.

F. Bars is supported by MTM2016-75980-P.

References

[1] Badr B., Bars F., and Lorenzo García E.; On twists of smooth plane curves, arXiv:1603.08711, Math. Comp. 88 (2019), 421-438, DOI: https://doi.org/10.1090/mcom/3317.10.1090/mcom/3317Search in Google Scholar

[2] Gille P., Szamuely T., Central simple algebras and Galois cohomology, Cambridge Studies in Advanced Mathematics 101, Cambridge University Press, 2006.10.1017/CBO9780511607219Search in Google Scholar

[3] Hanke T., The isomorphism problem for cyclic algebras and an application, ISSAC 2007, ACM, New York, 2007, pp. 181–186.10.1145/1277548.1277574Search in Google Scholar

[4] Hartshorne T., Algeraic Geometry, Springer Verlag, 1978, New York.Search in Google Scholar

[5] Jacobson N., Finite-dimensional division algebras over fields, Springer-Verlag, 1996.10.1007/978-3-642-02429-0Search in Google Scholar

[6] Jahnel J., The Brauer-Severi variety associated with a central simple algebra: a survey. See Linear Algebraic Groups and Related Structures 52, (2000), 1–60 or https://www.math.uni-bielefeld.de/lag/man/052.pdf.Search in Google Scholar

[7] Knus M.-A., Merkurjev A., Rost M., Tignol J.-P., The book of involutions, Colloquium Publications, 44, 1998, Providence, American Mathematical Society.10.1090/coll/044Search in Google Scholar

[8] Lorenzo García E., Twist of non-hyperelliptic curves, Rev. Mat. Iberoam. 33, 2017, no. 1, 169–182.10.4171/RMI/931Search in Google Scholar

[9] Lorenzo García E., Construction of Brauer-Severi varieties, arXiv:1706.10079, 2017.Search in Google Scholar

[10] Roé J., Xarles X., Galois descent for the gonality of curves, Arxiv:1405.5991v3,2015. To appear in Math. Research Letters.Search in Google Scholar

[11] Saltman D.J., Lectures on division algebras, Regional Conference Series in Mathematics, 94, 1999, Providence, RI: American Mathematical Society.10.1090/cbms/094Search in Google Scholar

[12] Serre J.P., Cohomologie Galoisianne, LNM 5, Springer, 1964.Search in Google Scholar

[13] Serre J.P., Local Fields, GTM, Springer, 1980.10.1007/978-1-4757-5673-9Search in Google Scholar

[14] Tengan E., Central Simple Algebras and the Brauer group, XVII Latin American Algebra Colloquium, 2009. See the book in http://www.icmc.usp.br/etengan/algebra/arquivos/cft.pdfSearch in Google Scholar

[15] Wedderburn J.H.M., On division algebras, Trans. Amer. Math. Soc. 22, 1921, no. 2, 129–135.10.1090/S0002-9947-1921-1501164-3Search in Google Scholar

Received: 2018-04-05

Accepted: 2018-09-14

Published Online: 2018-10-29

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https://doi.org/10.1515/math-2018-0101

Keywords for this article

Brauer-Severi varieties; Picard groups; Cyclic Algebras; Theory of Twists; Automorphisms

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