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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo


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Volume 16, Issue 1

Issues

Volume 13 (2015)

The Picard group of Brauer-Severi varieties

Eslam Badr / Francesc Bars / Elisa Lorenzo García
Published Online: 2018-10-29 | DOI: https://doi.org/10.1515/math-2018-0101

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

1

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 32 as in Theorem 2.5. A smooth model of B inside Pk9 is given by the intersectionτ∈Gal(L/k)τX where X/L is the variety in PL9 defined 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 (3r1)(3r2)2 over k. More precisely, it is k-isomorphic to a twist of the Fermat type curve X3r + arY3r + a2r Z3r = 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, PGLn+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 a k-isomorphism V×kk¯ΦV¯:=V×kk¯. The set of all twists of V modulo k-isomorphisms is denoted by Twistk (V), whereas the set of all twists Vof 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×kKΦV×kKξ(τ):=ΦτΦ1

where AutK(.) denotes the group of K-automorphisms of the object over K.

For K = k, the right hand side will be denoted by H1 (k, Autk(V)) or simply H1 (k, Aut(V)).

Definition 2.3

A Brauer-Severi variety B over k of dimension n is a twist of Pkn. 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 set BSnk is in bijection with Twistk(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 ak∗, 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):=LLeLe2. Multiplication is given by the relations: e.λ = σ (λ).e for λL, and e3 = a. The algebra (L/k, σ, a) is called the cyclic algebra associated to σ and the element ak∗.

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 akwhich 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, PGLn (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 FC (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(d1)(d2). 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 that C = C × kk is a smooth plane curve over k of degree d ≥ 4. Let Υk¯:C¯Pk¯2 be a morphism given by (the unique) gd2-linear system over k, then there exists a Brauer-Severi surface B defined over k, together with a k-morphism f:CB such that f×kk¯:C¯Pk¯2 is equal to Υk.

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 curve CPk¯2 over k of degree d ≥ 4, there exists a natural map

Σ:H1(k,Aut(C))H1(k,PGL3(k¯)),

moreover Σ1([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 Cto 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 akand r ∈ Z ≥ 1, the equation X3r + ar Y3r + a2r Z3r = 0, defines a smooth plane curve Car over 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

0Pic(B)Pic(B×kk¯)degZδBr(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 Cover k of the smooth plane curve Car, 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)=00a100010H1(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:Pk2Pk9 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)↦MτM−1, for some M ∈ GL10 (k). Consequently, (MVer3)(Pk2) is a model of B in Pk9, containing (MVer3)(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×kLPL2 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×kLPL2P9,

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

On the other hand, the twist ϕ:CCar 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 ∘ ΥLϕ × 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 ak∗, 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: eλ = σ (λ) ⋅ e for λL, and en + 1 = a. The algebra (L/k, σ, a) is called the cyclic algebra associated to σ and the element ak. It is trivial if and only if a is a norm of certain element of L. Its class in H1 (k, PGLn + 1 (k)) corresponds to the inflation of the cocycle in H1(Gal(L/k), PGLn + 1(L)) given by

ξ(σ)=000a10000100010.

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

Lemma 6.1

For any akand r ∈ ℤ≥1, the equation

i=0nairXi(n+1)r=0

defines a non-singular k-projective model Xar,n of degree (n + 1)r of a smooth projective variety inside Pkn, such that Aa:= [aXn:X0:…:Xn−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+1n1 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:PknPkm, 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 GLm + 1 (k) by [9] and since H1 (k, GLm + 1 (k)) = 1, then Ver~3([B]) is given in H1 (k, PGLm + 1 (k)) by τ ∈ Gal(k/k) ↦ MτM−1, for some M ∈ GLm + 1 (k). Consequently, (MVer3)(Pk2) is a model of B in Pk9, that contains (MVer3)(Ca1) inside, which is a twist of Xar,n over k associated to ξ : σAa.

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:PnPm:(X0:...:Xn)(ω0:...:ωm), where the ωk are equal to the products ωX0α0Xnαn=iXiαi with ∑iαi = n + 1 in alphabetical order. The automorphism of Xar,n as an automorphism of Vern(Xar,n) sends ωXinωXi+1n and ωXnnaωX1n.

Corollary 6.3

With the notation above,

Vern(X):i=0njli+jωXjnr=0B

Proof

By using [8, §3], we find that a matrix ϕ realizing the cocycle ξ, that is, ξ = ϕσϕ−1, sends ϕ(ωXin)=aijli+jωXjn and ϕ(ωXnn)=jlj1ωXjn. 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:PknPk2n+1n1 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 genus g=12(d1)(d2)3. The canonical embedding of C is isomorphic to the composition Ψ:CιPk2Verd3Pkg1, where ι comes from the (unique) gd2-linear system, all are defined over k. In particular, fixing a non-singular plane model FC (X, Y, ) = 0 in Pk2 of C, one may directly compute its canonical embedding into Pkg1 by applying the morphism Verd−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 Verd − 3, are Gal(k/k)-equivariant. Therefore, the natural maps

Aut(C¯)Aut(Pk¯2)=PGL3(k¯)Aut(Pk¯g1)=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 of Pk92r(r1) 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 model1 in ℙ9.

Proof

For non-hyperelliptic curves, see a description in [8], the canonical model gives a natural Gal(k/k)-inclusion Aut(C¯¯) ↪ PGLg (k), but we can go further, the action gives a Gal(k/k)-inclusion Aut(C¯¯) ↪ GLg (k). In this way, the natural map H1 (k, Aut(C¯¯)) → H1 (k, PGLg (k)), satisfies that the image of any 1-cocycle is equivalent to a 1-cocycle with values in GLg (k), and recall that H1 (k, GLg (k)) is trivial by applying Hilbert’s Theorem 90. This allows us to compute equations for twists via change of variables in GLg (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 Vd − 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.

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Footnotes

  • 1

    In [9] the equations for Brauer-Severi surfaces are also obtained in ℙ9. 

About the article

Received: 2018-04-05

Accepted: 2018-09-14

Published Online: 2018-10-29


Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1196–1203, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0101.

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© 2018 Badr et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0

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