ACTA ARITHMETICA Online First version Hypersurface model-fields of definition for smooth hypersurfaces and their twists by Eslam Badr (Cairo) and Francesc Bars (Barcelona) 1. Introduction. Let X0 , . . . , Xn be a homogeneous coordinate system for Pkn , the n-dimensional projective space over k. Given a smooth projective variety V ⊂ Pnk , the group of birational transformations of V onto itself is denoted by Bir(V ), the group of automorphisms of V (that is, the group of biregular transformations of V onto itself) is Aut(V ), and we denote by Lin(V ) the subgroup of automorphisms of V induced by projective lin- ear transformations in Aut(Pkn ) = PGLn+1 (k), the general linear group of (n + 1) × (n + 1) projective matrices. Now, let V be a smooth hypersurface in Pnk , that is, an (n−1)-dimensional smooth projective variety identified with a hypersurface model HV ,d,n repre- sented by a single homogeneous polynomial equation, say F (X0 , . . . , Xn ) = 0 of some degree d over k without singularities (assume once and for all that d ≥ 4). It is known that a smooth plane curve V of degree d ≥ 4 has finitely many automorphisms, and moreover any automorphism is induced by a pro- jective linear transformation of Pk2 , thus Aut(V ) = Lin(V ). Matsumura and Monsky (1946) showed that, for n ≥ 3, Lin(V ) is a finite group and moreover Aut(V ) = Lin(V ) except possibly when (n, d) = (3, 4) (see [6, Theorems 1, 2]). Definition 1.1. A smooth projective variety V defined over a field k is called a smooth L-hypersurface over k of degree d in Pnk and L is a hypersur- face model-field of definition for V , where L/k is a field extension inside k, if the base extension V ⊗k L is L-isomorphic to a non-singular hypersurface 2020 Mathematics Subject Classification: Primary 14J70, 11D72, 11D41; Secondary 14J50, 11G99, 14J20. Key words and phrases: fields of definition, hypersurface models, twists, automorphism groups. Received 24 May 2018; revised 15 July 2019. Published online *. F. Bars was supported by MTM2016-75980-P. DOI: 10.4064/aa180524-31-7 [1] c Instytut Matematyczny PAN, *** 2 E. Badr and F. Bars model HV ⊗k L,d,n : FV ⊗k L (X0 , . . . , Xn ) = 0 of degree d with coefficients in L. For the special case L = k, V is simply called a smooth hypersurface over k. Suppose that V is a smooth k-hypersurface over k of degree d ≥ 4 in Pkn . Hence, by finiteness and linearity of Aut(V ⊗k k) for (n, d) 6= (3, 4), we find that V := V ⊗k k has a linear series, which allows us to embed Υ : V ,→ Pnk as a smooth hypersurface. Lemma 1.2. The linear system Υ for n, d ≥ 3 such that (n, d) 6= (3, 4), is unique up to PGLn+1 (k)-conjugation. Proof. Suppose that HV ,d,n and HV0 ,d,n are two non-singular hypersurface models for V in Pnk of degree d. Then, both models are k-isomorphic via an isomorphism, say φ. The same argument in the proof of [6, Theorem 2] assures that φ∗ maps hyperplane sections on HV ,d,n to hyperplane sections on HV0 ,d,n , and so φ is induced by a linear transformation of Pnk (for n ≥ 4 such an invari- ance property on hyperplane sections follows from a theorem of Grothendieck– Lefschetz or following [6] a theorem of Severi–Lefschetz–Andreotti). Thus, for (n, d) 6= (3, 4), we can always think about Aut(V ) as a finite subgroup of PGLn+1 (k), leaving invariant a fixed non-singular hypersurface model HV ,d,n : FV (X0 , . . . , Xn ) = 0 of degree d coming from the embedding Υ : V ,→ Pnk over k. In other words, any other non-singular hypersurface model over k is defined by an equation of the form FP −1 V (X0 , . . . , Xn ) := FV (P (X0 , . . . , Xn )) = 0 for some P ∈ PGLn+1 (k). The aim of this paper is to make a study for fields of definition of non-singular hypersurface models of a smooth k-hypersurface V over k, also for its twists, by considering the embedding Aut(V ) ,→ PGLn+1 (k). We note that if the smooth projective variety V , or any of its twists over k, is a smooth hypersurface over k, then we have an embedding of Gal(k/k)-groups for its automorphism group into PGLn+1 (k). This approach leads to two natural questions: first, given a smooth projective variety V defined over a field k and admitting a non-singular k-hypersurface model, does it have a non-singular hypersurface model over k; and secondly, if the answer is yes, does every twist of V over k also have a non-singular hypersurface model over k? For both questions the answer is No: in general, it does not. We obtain results for the varieties for which the above questions always have an affirmative answer, and we show different examples concerning the negative general answer. The paper is a generalization of our joint work [1] with Elisa Lorenzo García, where the same problem was addressed, but for smooth hypersurfaces in P2 , that is, for smooth plane curve of degree d ≥ 4. Smooth hypersurfaces and their twists 3 2. Statements of the results. First, we study the minimal field L where there exists a non-singular model over L for a smooth k-hypersur- face V defined over k. We show the following, which follows from [7]. Theorem 2.1. Let V be a smooth k-hypersurface over a perfect field k of degree d ≥ 4 in Pnk such that (n, d) 6= (3, 4). Then V is not necessar- ily a smooth hypersurface over k. However, it is so in any of the following situations: (i) if V has k-rational points, i.e. when V (k) 6= ∅, (ii) if gcd(d, n + 1) = 1, (iii) if the (n + 1)-torsion Br(k)[n + 1] of the Brauer group Br(k) is trivial. In general, V has a non-singular hypersurface model over a field extension L/k of degree [L : k] | n + 1. Also, in the case of number fields k, we show in §4.1 an example of a smooth k-hypersurface over k, which is not a smooth hypersurface over k, but it does over a Galois field extension of degree n + 1 over k. Notation and conventions. We write Gal(L/k) for the Galois group of the extension L/k, and we consider left actions. The Galois cohomology sets of a Gal(L/k)-group G when L/k is Galois are denoted by Hi (Gal(L/k), G) with i ∈ {0, 1} respectively. For the particular case L = k, we use Gk instead of Gal(k/k) and H1 (k, G) instead of H1 (Gal(k/k), G). Second, we assume that V is a smooth hypersurface over k. We obtain the next theorem characterizing the twists of V , which are also smooth hy- persurfaces over k. In particular, we prove the following. Theorem 2.2. Let V be a smooth hypersurface over a perfect field k iden- tified with a fixed non-singular hypersurface model HV,d,n : FV (X0 , . . . , Xn ) = 0, where FV (X0 , . . . , Xn ) ∈ k[X0 , . . . , Xn ] with (n, d) 6= (3, 4). Then there exists a natural map Σ : H1 (k, Aut(V )) → H1 (k, PGLn+1 (k)), defined by the inclusion Aut(HV,d,n ⊗k k) ⊆ PGLn+1 (k) as Gk -groups. The preimage Σ −1 ([Pnk ]) is formed by the set of all twists of V over k that are smooth hypersurfaces over k, where [Pnk ] denotes the class of the triv- ial Brauer–Severi variety of dimension n over k. Moreover, any such twist is obtained through an automorphism of Pkn , that is, the twist is k-isomorphic to HFM −1 V,d,n : FM −1 V (X0 , . . . , Xn ) := FV (M (X0 , . . . , Xn )) ∈ k[X0 , . . . , Xn ], for some M ∈ PGLn+1 (k). 4 E. Badr and F. Bars We can reinterpret the map Σ as the map sending a twist V 0 over k to the Brauer–Severi variety B where it lives (cf. [7, Lemma 5]). Then, we have assertions similar to those in Theorem 2.1. Corollary 2.3. Let V be a smooth hypersurface over a perfect field k of degree d ≥ 4 inside Pnk with (n, d) 6= (3, 4). The map Σ in Theorem 2.2 is trivial if gcd(d, n + 1) = 1 or Br(k)[n + 1] is trivial. In particular, for such situations, any twist V 0 for V over k is also a smooth hypersurface over k. On the other hand, we construct in §4.4 a one-parameter family HFa ,d=2p,n , for a ∈ k, of smooth hypersurfaces over a number field kd where p is an odd prime integer, and we show that each member of the family has a twist over kd that does not admit a hypersurface model over kd . Finally, we study the twists for a smooth hypersurface V over k such that Aut(V ) is a cyclic group. Definition 2.4. Let V /k : FV (X0 , . . . , Xn ) = 0 be a smooth hypersur- face over a perfect field k, where FV (X0 , . . . , Xn ) ∈ k[X0 , . . . , Xn ]. We call a twist V 0 for V over k a diagonal twist if there exists an M ∈ PGLn+1 (k) and a diagonal matrix D ∈ PGLn+1 (k) such that V 0 is k-isomorphic to F(M D)−1 V (X0 , . . . , Xn ) := FV (M D(X0 , . . . , Xn )) = 0, where F(M D)−1 V (X0 , . . . , Xn ) ∈ k[X0 , . . . , Xn ]. We prove the following. Theorem 2.5 (Diagonal twists). Let V /k : FV (X0 , . . . , Xn ) = 0 be a smooth hypersurface over a perfect field k of degree d ≥ 4 with (n, d) 6= (3, 4). Assume that Aut(V ⊗k k) ,→ PGLn+1 (k), given by the linear system, is a non-trivial cyclic group of order m generated by ψ = diag(1, ζm a1 , . . . , ζ an ) m for some ai ∈ N, where ζm denotes a fixed primitive mth root of unity and m is coprime to the characteristic of k. Then all the twists in Twistk (V ) are diagonal over k given by a non-singular polynomial equation FD−1 V (X0 , . . . , Xn ) = 0 where FD−1 V (X0 , . . . , Xn ) ∈ k[X0 , . . . , Xn ] and D is a diagonal matrix in PGLn+1 (k). In particular, the map Σ in Theorem 2.2 is trivial. Remark 2.6. Let V /k be a smooth hypersurface over a perfect field k of characteristic p ≥ 0, and identify it with a non-singular hypersurface model HV,d,n : FV (X0 , . . . , Xn ) = 0 over k with d ≥ 4 and (n, d) 6= (3, 4). Sup- pose also that Aut(HV,d,n ⊗k k) ⊆ PGLn+1 (k) is a cyclic group of order n, generated by a matrix ψ whose conjugacy class in PGLn+1 (k) does not con- tain elements of diagonal shapes. Then the twists of V over k whose image under Σ is trivial (i.e., the ones that are smooth hypersurfaces over k) are expected not to be represented by diagonal twists. For example, for n = 2 Smooth hypersurfaces and their twists 5 in [1], we provide a smooth plane curve with a cyclic non-diagonal auto- morphism group in the above sense, where not all of its twists are diagonal. Constructing such examples in higher-dimensional Pn requires knowledge of the structure of automorphism groups and also the twisting theory for smooth hypersurfaces living there. 3. Brauer–Severi varieties and central simple algebras. Let U be a quasi-projective variety defined over a perfect field k. A twist for U is a variety U 0 defined over k that is isomorphic over k to U , but not necessarily over k. A twist U 0 is called trivial if U and U 0 are k-isomorphic. The set of all twists of U modulo k-isomorphisms is denoted by Twistk (U ). It is well known that the set Twistk (U ) is in one-to-one correspondence with the first Galois cohomology set H1 (k, Aut(U ⊗k k)) given by [U 0 ] 7→ ξ : τ 7→ ξτ := φ ◦ τ φ−1 for τ ∈ Gk , where φ : U 0 ⊗k k → U ⊗k k is a fixed k-isomorphism (see [9, Chap. III]). Brauer–Severi varieties. A Brauer–Severi variety B of dimension n over a perfect field k is a twist of the n-dimensional projective space Pnk over k. A field extension L/k is said to be a splitting field of B if B⊗k L ' PnL , and we say that L/k splits B or that B splits over L. In particular, we obtain Corollary 3.1. The set Twistk (Pnk ) is in bijection with H1 (k, Aut(Pnk )) = H1 (k, PGLn+1 (k)). Moreover, we have Theorem 3.2 (Severi, Châtelet, Lichtenbaum). Let B be a Brauer–Severi variety of dimension n over a perfect field k. Then there exists a field ex- tension L/k of degree [L : k] | n + 1 such that L splits B. Moreover, B splits over k if it has k-rational points or contains a hypersurface of degree rela- tively prime with n + 1. Proof. The result is due to Severi (cf. [8, X, §6, Exercise 1] in the original French edition), Châtelet (cf. his PhD thesis [2]), and Lichtenbaum (cf. [4, Theorem 5.4.10]). One can also read the proof of [7, Theorem 5]. We deduce from [7, Lemma 4] the following. Theorem 3.3 (Roé–Xarles). Let V be a smooth projective variety over a perfect field k. Suppose that, for some fixed n ≥ 2, there is a unique (modulo automorphisms) n-dimensional linear series over k invariant under the Gk - action, giving a morphism h : V ⊗k k → Pnk . Then there exists a Brauer– Severi variety B of dimension n defined over k, together with a k-morphism g : V ,→ B, such that g ⊗k k : V ⊗k k → Pnk is equal to h. 6 E. Badr and F. Bars Central simple algebras. A central simple algebra over a field k is a finite-dimensional associative algebra over k which is simple, i.e. it con- tains no non-trivial (two-sided) ideal and the multiplication operation is not uniformly zero, and for which the center is exactly k. A field extension L/k is said to be a splitting field of a central simple algebra A over k if A ⊗k L ' Mn (L) for some n, and we say that L/k splits A. Example 3.4 (Cyclic algebras). Let L/k be a cyclic extension of degree n + 1 with Gal(L/k) = hσi. Then an element of H1 (Gal(L/k), PGLn+1 (L)) represented by a 1-cocycle f : Gal(L/k) → PGLn+1 (L) is completely deter- mined by the value of f (σ), which is subject to σ2 σn f (σ) · σ (f (σ)) · (f (σ)) · . . . · (f (σ)) = 1. For instance, let a ∈ k ∗ and consider the matrices     0 1 0 ... 0 0 0 0 ... 0 a 0 0 1 . . . 0 0     1 0 . . . 0 0     0 0 0 . . . 0 0 0 1 . . . 0 0 Ca :=   , Da :=  . . .   .   .. ..   ..    0 0 . . . . . . 0 1     0 0 ... 1 0 a 0 ... ... 0 0 Define a 1-cocycle f by setting f (σ) = Ca mod L∗ . Hence σ2 σn f (σ) · σ (f (σ)) · (f (σ)) · . . . · (f (σ)) = Can+1 mod L∗ = I mod L∗ , where I is the identity matrix. According to [11, Theorem 5.4], we can asso- ciate to this 1-cocycle a central simple algebra A over k of dimension (n+1)2 that splits over L, by considering the set of matrices M ∈ Mn+1 (L) satisfying Ca σM Ca−1 = M . One finds that I, Ca , . . . , Can ∈ A as well as Sb := diag(b, σ(b), . . . , σ n (b)) for b ∈ L. Sb Cai is a k-subalgebra of the correct dimension (n + 1)2 , and L Therefore, corresponds to the algebra A defined by the 1-cocycle f . This kind of k- algebras is also known as the cyclic algebra (χ, a) associated to a ∈ k ∗ and ' the character χ : Gal(L/k) → Z/(n + 1)Z defined by χ(σ) = −1 mod (n + 1). In the above computations, we may replace Ca with Da and we get symmetrically the cyclic algebra (χ, a) associated to a ∈ k ∗ and the character ' −1 χ : Gal(L/k) → Z/(n+1)Z sending σ to 1 mod (n+1), since Ca Sb = σ Sb Ca is changed to Da Sb = σSb Da . For complete details, we refer to [11, Example 5.5]. Smooth hypersurfaces and their twists 7 Theorem 3.5 (J. H. Maclagan-Wedderburn, R. Brauer). Given a central simple algebra A over k, there exists a unique (up to isomorphism) division algebra D with center k and a positive integer n such that A is isomorphic to Mn (D). Consequently, the dimension dimk (A) of A over k is always a square. Theorem 3.5 gives a strict relation between central simple algebras and division algebras, and suggests the introduction of the following equivalence relation: Two central simple algebras A1 and A2 over the same field k are equivalent if there are positive integers m, n such that Mm (A1 ) ' Mn (A2 ). Equivalently, A1 and A2 are equivalent if A1 and A2 are (up to isomorphism) matrix algebras over a division algebra. Definition 3.6. The set of all Brauer equivalence classes of central simple algebras over k equipped with the tensor product of k-algebras is an abelian group (cf. [4, Proposition 2.4.8]), known as the Brauer group of k and denoted by Br(k). The period of a central simple algebra over k is defined to be its order as an element of the Brauer group. The m-torsion Br(k)[m] of the Brauer group Br(k) is the set of all elements of Br(k) of order dividing m. Recall that each Brauer equivalence class contains a unique (up to iso- morphism) division algebra. Define the index of a central simple algebra to be the degree of the division algebra D that is Brauer equivalent to it, i.e. the square root of the dimension of D over k. In particular we have Corollary 3.7 (cf. [3]). The period of a central simple algebra over k divides its index, and hence is finite. Interplay. In the literature we find several approaches to the connection between central simple algebras and Brauer–Severi varieties; first, the con- nection between quaternion algebras and plane conics observed by E. Witt in [13]. In its general form, we mention for example the most elementary one promoted by J.-P. Serre in his books [8, 9]. The main observation is that central simple algebras of dimension (n + 1)2 over a perfect field k as well as n-dimensional Brauer–Severi varieties over k can both be described by classes in one and the same cohomology set H1 (k, PGLn+1 (k)). 4. Proofs of the results 4.1. A smooth k-hypersurface without a non-singular hypersur- face model over k. Fix an algebraic closure Q of Q, and let k ⊂ Q be a 8 E. Badr and F. Bars number field. Suppose that n Y f (t) = tn+1 + λn tn + · · · + λ1 t + (−1)n+1 λ0 = (t − ai ) ∈ k[t] i=0 is an irreducible polynomial of degree n + 1 ≥ 3 over k[t] whose splitting field kf over k satisfies Gal(kf /k) ∼ = Z/(n + 1)Z. Such a polynomial exists since the inverse Galois problem for the cyclic group Z/(n + 1)Z is solvable (recall that any solvable group is realizable as a Galois group over a number field by the contribution of Shafarevich in 1954). Fix a generator σ : a0 → a1 → a2 → · · · → an−1 → an → a0 for the Galois group Gal(kf /k). Take a positive integer d not relatively prime to n + 1 such that the following Qn holds: there exists β ∈ k ∗ which is not a n+1 d norm in kf and λ0 β = i=0 σ (α), equivalently β d = Nkf /k (α/λ0 ), for i some α ∈ kf∗ \ k ∗ . Next, we define a smooth hypersurface over kf by the equation n  i−1  d X 1 Y j Hf,α,d,n : λ0 X0 + i−1 σ (α) Xid = 0 i=1 λ0 j=0 of degree d ≥ 4 such that (n, d) 6= (3, 4). The matrix   0 0 ... 0 β   1 0 . . . 0 0    φσ := 0 1 . . . 0 0    .   ..    0 0 ... 1 0 defines an isomorphism φσ = Cβ : σ Hf,α,d,n → Hf,α,d , which satisfies Weil’s condition of descent [12] (φσn+1 = φn+1 σ = 1). We therefore find that the variety is defined over k, and that there exists an isomorphism ϕ0 : Vk → Hf,α,d,n where Vk is a rational model such that ψσ = φ−1 σ = σ −1 τ −1 ϕ0 ◦ ϕ0 ∈ PGLn+1 (k). The formula ψτ := ϕ0 ◦ ϕ0 defines an element of H1 (Gal(kf /k), PGLn+1 (kf )), corresponding to a cyclic algebra which is non-trivial because β is not a norm of an element of kf (cf. [5, §2.1]). Conse- quently, ϕ0 is not given by an element of PGLn+1 (kf ), or even PGLn+1 (k f ), since its cohomology class after applying the inflation map is not trivial. Therefore, the variety Vk is not a smooth hypersurface over k (otherwise, Vk is identified via a k-isomorphism ψ0 with a non-singular hypersurface model defined over k. Thus, by [6, Theorems 1, 2], the cohomology class [ϕ0 ◦ ψ0 ] is represented by an M ∈ PGLn+1 (k f ), which it is not since [ϕ0 ] 6= 1). As a concrete example, we specify the above construction in P2 , P3 and P4 . Smooth hypersurfaces and their twists 9 (1) In P2 : take k = Q and consider the irreducible polynomial f (t) = t3 + 12t2 − 64 over k (thus λ0 = 64). As we can check with SAGE [10], the discriminant of the field kf is a power of 3, and the prime 2 becomes inert in kf , hence is not a norm in kf . Consequently, we can assume, for example, that d = 9m − 12, β = 2 and α = a0 8m with m ≥ 2 an integer. (2) In P3 : take k = Q and consider the irreducible polynomial f (t) = t4 + t3 + 2t2 − 4t + 3 over k (thus λ0 = 3). Then kf is a cyclic extension of Q with Galois group isomorphic to Z/4Z (we can think of it inside Q(ζ13 )), moreover, λ0 = 3 splits completely in kf , so it may be a norm. In the ring of integers Okf of kf , β = 17 = p1 p2 , where p1 and p2 are certain prime ideals of Okf , also one of the generators of p1 as a fractional ideal, say γ, belongs to kf \ Q. It is easy to check that Nkf /Q (γ) = 172 , and we can choose α := 3 · 17m γ ∈ kf \ Q with m ≥ 1, giving degrees d = 4m + 2 (using Sage: K.=NumberField(x^4+x^3+2*x^2-4*x+3); K.ideal(17).factor(); >(Fractional ideal(1/3*a^3-1/3*a+3)) *(Fractional ideal(1/3*a^3-1/3*a-6)); suggests taking γ = 31 a3 − 31 a + 3). (3) In P4 : the polynomial f (t) = t5 − t4 − 4t3 + 3t2 + 3t − 1 is irreducible over Q. Its splitting field Qf is cyclic over Q of degree 5 (we can think Qf = Q(cos(2π/11)) ⊆ Q(ζ11 )). In the ring of integers OQf , the torsion units are ±1, and the roots ai of f are units in OQf . Suppose that m > 1 is an integer satisfying gcd(m, n + 1 = 5) = 1 and gcd(ϕ(m), 5) = 1, where ϕ is the Euler function. Now, take d = 5m and k = Q(ζd ) with ζd a fixed dth primitive root of unity inside Q. Note that f (t) is irreducible over k = Q(ζd ), since k ∩ Qf = Q. Also, kf does not contain torsion roots of unity other than hζd i, therefore ζd is not a norm from kf to k. In particular, we can set β = ζd and α = a0 . Corollary 4.1. Let V be a smooth k-hypersurface over a perfect field k of degree d ≥ 4 in Pkn such that (n, d) 6= (3, 4). Then V is not necessarily a smooth hypersurface over k. 4.2. Minimal fields of definition for non-singular hypersurface models. In this subsection, V is a smooth k-hypersurface of degree d ≥ 4 in Pnk with n ≥ 2 and (n, d) 6= (3, 4). Accordingly, V := V ⊗k k has an n-dimensional linear series over k that allows us to embed Υ : V ,→ Pnk as a non-singular hypersurface, and such a linear series is unique modulo conjugation in PGLn+1 (k). We first show 10 E. Badr and F. Bars Proposition 4.2. Let V be a smooth k-hypersurface over k of degree d ≥ 4 in Pkn , where n ≥ 2 and (n, d) 6= (3, 4). There exists a non-singular hypersurface model over a field extension L/k of degree [L : k] | n + 1. Proof. We know from [7, Lemma 4] that the set of k-morphisms (modulo automorphisms) to some n-dimensional Brauer–Severi variety over k are in bijection with the base-point free n-dimensional linear series over k which are invariant under the Gk -action. Consequently, we may apply Theorem 3.3 to obtain a k-morphism g : V ,→ B to a Brauer–Severi variety B of dimension n over k such that g ⊗k k : V ,→ Pkn equals to Υ . Moreover, by [7, Theorem 13(5)], there exists a field extension L/k of index [L : k] | n + 1 that splits B (this means that B ⊗k L is L-isomorphic to PnL ). Hence, we reduce to an em- bedding of V ⊗k L into PnL as the smooth variety g(V ) ⊗k L. By assumption, g(V ) ⊗k k is a hypersurface inside Pkn ; then so does g(V ) ⊗k L ⊂ PnL . Con- sequently, g(V ) ⊗k L has dimension n − 1 and therefore it is a non-singular hypersurface model for V ⊗k L over L. Second, we show Proposition 4.3.Let V be a smooth k-hypersurface over k of degree d ≥ 4 in Pkn , where n ≥ 2 and (n, d) 6= (3, 4). Then V is a smooth hypersurface over k if V (k) 6= ∅, gcd(d, n + 1) = 1, or Br(k)[n + 1] is trivial. Proof. Using [7, Theorems 13(1),(2)], we find that the base field k splits B when V (k) 6= ∅ or gcd(d, n + 1) = 1. On the other hand, let [A] be the image of B in the Brauer group Br(k) (in particular, B splits over a field extension L/k if and only if A does over L/k). Due to Châtelet’s thesis [2, Chapter IV.1], the division algebra associated to A has dimension dividing n + 1. That is, B corresponds to an element of the (n + 1)-torsion of Br(k), since the order of a central simple algebra as an element of Br(k) divides its index, which is the square root of the dimension of the associated division algebra (see Corollary 3.7). Therefore, B also splits over k if Br(k)[n + 1] is trivial, being associated to a trivial central simple algebra over k. By the above discussion, we can see that our variety V lives inside a trivial Brauer–Severi variety in any of the prescribed situations. This in turn gives a non-singular hypersurface model over k, and we conclude. Corollary 4.4. Let V be a smooth k-hypersurface over k of degree d ≥ 4 in Pkn , where n ≥ 2 and (n, d) 6= (3, 4). Then V is a smooth hypersurface over k if (i) k is an algebraically closed field; (ii) k is a finite field; (iii) k is the function field of an algebraic curve over an algebraically closed field; Smooth hypersurfaces and their twists 11 (iv) k is an algebraic extension of Q, containing all roots of unity; or (v) k = R and n + 1 odd. Proof. In the following cases, every division algebra over a field k is k itself, so that the Brauer group Br(k) is trivial: • k is an algebraically closed field (cf. [11, Example 4.2]). • k is a finite field (Wedderburn’s Little Theorem, cf. [8, p. 162]). • k is the function field of an algebraic curve over an algebraically closed field (Tsen’s Theorem, cf. [4, Theorem 6.2.8]). More generally, the Brauer group vanishes for any quasi-algebraically closed field. • k is an algebraic extension of Q, containing all roots of unity (cf. [8, p. 162]). Finally, there are just two non-isomorphic real division algebras with cen- ter R: R itself and the quaternion algebra H. Since H ⊗ H ' M4 (R), the class of H has order two in the Brauer group. That is, the Brauer group Br(R) is the cyclic group of order two and then, for n + 1 odd, Br(R)[n + 1] does not contain non-trivial elements of order dividing n + 1. Now, the result is an immediate consequence of Proposition 4.3. 4.3. Twists which are smooth hypersurfaces over the base field k. Now, assume that V is a smooth hypersurface over k, in particular V is k-iso- morphic to a non-singular hypersurface model of degree d ≥ 4 of the form HFV ,d,n : FV (X0 , . . . , Xn ) := FM −1 V (X0 , . . . , Xn ) = FV (M (X0 , . . . , Xn )) ∈ k[X0 , . . . , Xn ] for some M ∈ PGLn+1 (k) with n ≥ 2 and (n, d) 6= (3, 4). Since we have Aut(HFV ,d,n ⊗k k) ⊂ PGLn+1 (k) as Gk -groups, Aut(V ) is naturally embed- ded into PGLn+1 (k) as Gk -groups, and we get a well-defined map Σ : Twistk (V ) = H1 (k, Aut(V )) → H1 (k, PGLn+1 (k)). Proof of Theorem 2.2 and Corollary 2.3. Let [Pnk ] denote the class of the trivial Brauer–Severi variety in Twistk (Pnk ) = H1 (k, PGLn+1 (k)). If a twist V 0 /k is k-isomorphic to a non-singular hypersurface model FV 0 (X0 , . . . , Xn ) = 0 over k, then FV 0 (X0 , . . . , Xn ) = 0 and FV (X0 , . . . , Xn ) = 0 are isomor- phic through some M 0 ∈ PGLn+1 (k) by [6, Theorem 1, 2] when n ≥ 3 and it is well known when n = 2 in the case of plane curves of degree ≥ 4. Hence, the corresponding 1-cocycle σ 7→ M 0 ◦ σ (M 0 )−1 ∈ Aut(HFV ,d,n ⊗k k) is triv- ial in H1 (k, PGLn+1 (k)), being cohomologous to the trivial 1-cocycle. Con- versely, if Σ([V 0 ]) is trivial (that is, V 0 /k lives inside a trivial Brauer–Severi variety of dimension n over k), then it must be given by a k-isomorphism ϕ : {FV (X0 , . . . , Xn ) = 0} → V 0 induced by some M̃ ∈ PGLn+1 (k), i.e. V 0 is k-isomorphic to F(M̃ )−1 V (X0 , . . . , Xn ) = 0. This would give a non-singular hypersurface model over k for V 0 . 12 E. Badr and F. Bars In general, any twist V 0 for V over k is again a smooth k-hypersurface over k in Pnk of degree d. It only remains to apply Theorem 2.1 for V 0 to conclude that V 0 is a smooth hypersurface over k if gcd(d, n + 1) = 1 or if Br(k)[n + 1] is trivial. In particular, Σ is the trivial map in both cases, which was to be shown. 4.4. A smooth hypersurface over a number field k, having a twist which is not a smooth hypersurface over k. Let p be an odd prime number and fix ζp , a primitive pth root of unity inside Q. Assume that HF,d,n : F (X0 , . . . , Xn ) = 0 ⊂ Pnk is a smooth hypersurface of degree d = 2p, not relatively prime to n+1, defined over the number field k = Q(ζp ). Suppose further that the projective matrix   0 0 . . . 0 ζp   1 0 . . . 0 0    φ := 0 1 . . . 0 0    .   ..    0 0 ... 1 0 induces an automorphism of HF,d=2p,n ⊗k k. For example, consider the fol- lowing family parametrized by a ∈ k: n n−1 n aXip Xjp = 0. X X X HFa ,d=2p,n : Fa (X0 , . . . , Xn ) = Xid + i=0 i=0 j=i+1 If m ∈ k∗\ (k ∗ )p , then xp − m is irreducible √ in k[x] by a theorem of Abel, √m = k( m) has Galois group Gal(L/k) = hσi ' p the Galois field extension √ L Z/pZ, where σ( p m) = ζp p m. Define the 1-cocycle by ξ : σ 7→ φ ∈ H1 (Gal(Lm /k), PGLn+1 (Lm )). Since no new primitive root of unity appears in Lm other than ζp , ζp is not a norm in Lm , [ξ] is non-trivial in H1 (Gal(Lm /k), PGLn+1 (Lm )) by [5, §1.1], and hence the image of ξ in H1 (k, PGLn+1 (k)), which coincides with the inflation of σ 7→ φ, is not trivial. Consequently, it corresponds to a twist V 0 for HFa ,d,n over k living inside a non-trivial Brauer–Severi variety of dimension n over k (that is, Σ([V 0 ]) 6= [Pnk ]). 4.5. Diagonal twists. Let V /k : FV (X0 , . . . , Xn ) = 0 be a smooth hypersurface over a perfect field k. Assume that Aut(V ⊗k k) ,→ PGLn+1 (k) is a non-trivial cyclic group of order m (relatively prime to the characteristic of k), generated by ψ = diag(1, ζm a1 , . . . , ζ an ) for some a ∈ N, where ζ is a m i m fixed primitive mth root of unity in k. Smooth hypersurfaces and their twists 13 Proof of Theorems 2.5 and 2.6. It suffices to notice that the embedding Aut(V ⊗k k) ,→ PGLn+1 (k) factors through GLn+1 (k). Thus the map Σ in Theorem 2.2 factors as follows: Σ : H1 (k, Aut(V ⊗k k)) → H1 (k, GLn+1 (k)) → H1 (k, PGLn+1 (k)). Moreover, H1 (k, GLn+1 (k)) = 1 by Hilbert’s 90th Theorem, so the map Σ is trivial. By Theorem 2.2, any twist has a non-singular plane model of the form FP −1 V (X0 , . . . , Xn ) = 0 over k, for some P ∈ PGLn+1 (k). Since P ◦ σ (P −1 ) ∈ Aut(V ⊗k k) = hdiag(1, ζm a1 , . . . , ζ an )i for any σ ∈ G , we m k σ have P = P ◦ diag(1, v1 , . . . , vn ) for some mth roots of unity vi . Writing P = (ai,j ), one easily deduces that σ(ai,j ) = vj ai,j for all i, j. Hence, for any fixed integer j, we have σ(ai,j )a−1 −1 −1 i,j = σ(ai0 ,j )ai0 ,j . That is, ai,j ai0 ,j is Gk -invariant, which in turn gives ai,j = mi ai0 ,j for some mi ∈ k. In par- ticular, P reduces to M D for some diagonal projective (n + 1) × (n + 1) matrix D and M ∈ PGLn+1 (k). This proves that all the twists are diagonal. However, the non-singular hypersurface model F(M D)−1 V (X0 , . . . , Xn ) = 0 over k is k-isomorphic through M to FD−1 V (X0 , . . . , Xn ) = 0. Consequently, FD−1 V (X0 , . . . , Xn ) = 0 defines a non-singular hypersurface model over k for the twist. Acknowledgements. The authors would like to express their gratitude to David Kohel who suggested the generalization of the work in [1] to hyper- surfaces instead of plane curves, and also to Joaquim Roé and Xavier Xarles for their useful comments and suggestions. We thank the referee for numer- ous helpful comments on the paper, both large and small, that improved the paper to its present form. References [1] E. Badr, F. Bars, and E. Lorenzo García, On twists of smooth plane curves, Math. Comp. 88 (2019), 421–438. [2] F. Châtelet, Variations sur un thème de H. Poincaré, Ann. Sci. École Norm. Sup. 61 (1944), 249–300. [3] A. J. de Jong, The period-index problem for the Brauer group of an algebraic surface, Duke Math. J. 123 (2004), 71–94. [4] P. Gille and T. Szamuely, Central Simple Algebras and Galois Cohomology, Cam- bridge Stud. Adv. Math. 101, Cambridge Univ. Press, Cambridge, 2006. [5] T. Hanke, The isomorphism problem for cyclic algebras and an application, in: ISSAC 2007, ACM, New York, 2007, 181–186. [6] H. Matsumura and P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ. 3 (1963), 347–361. [7] J. Roé and X. Xarles, Galois descent for the gonality of curves, Math. Res. Lett. 25 (2018), 1567–1589. [8] J.-P. Serre, Local Fields, Grad. Texts in Math. 67, Springer, New York, 1979. 14 E. Badr and F. Bars [9] J.-P. Serre, Cohomologie galoisienne, 5th ed., Lecture Notes in Math. 5, Springer, Berlin, 1994. [10] W. Stein et al., Sage mathematics software (version 4.3.4), the Sage Development team, 2010, http://www.sagemath.org. [11] E. Tengan, Central simple algebras and the Brauer group, in: XVII Latin American Algebra Colloquium, 2009. [12] A. Weil, The field of definition of a variety, Amer. J. Math. 78 (1956), 509–524. [13] E. Witt, Über ein Gegenbeispiel zum Normensatz, Math. Z. 39 (1935), 462–467. Eslam Badr Francesc Bars Department of Mathematics Departament de Matemàtiques Faculty of Science Universitat Autònoma de Barcelona Cairo University 08193 Bellaterra, Catalonia 12613 Giza, Egypt E-mail: francesc@mat.uab.cat E-mail: eslam@sci.cu.edu.eg Abstract (will appear on the journal’s web site only) Given a smooth projective variety of dimension n − 1 ≥ 1 defined over a perfect field k that admits a non-singular hypersurface model in Pnk over k, a fixed algebraic closure of k, it does not necessarily have a non-singular hyper- surface model defined over the base field k. We first show an example of such phenomenon: a variety defined over k admitting non-singular hypersurface models but none defined over k. We also determine under which conditions a non-singular hypersurface model over k may exist. Now, even assuming that such a smooth hypersurface model exists, we wonder about the exis- tence of non-singular hypersurface models over k for its twists. We introduce a criterion to characterize twists possessing such models and we also show an example of a twist not admitting any non-singular hypersurface model over k, i.e. for any n ≥ 2, there is a smooth projective variety of dimension n − 1 over k which is a twist of a smooth hypersurface variety over k, but itself does not admit any non-singular hypersurface model over k. Finally, we obtain a theoretical result to describe all the twists of smooth hypersur- faces with cyclic automorphism group having a model defined over k whose automorphism group is generated by a diagonal matrix. The particular case n = 2 for smooth plane curves was studied by the authors jointly with E. Lorenzo García in [Math. Comp. 88 (2019)], and we deal here with the problem in higher dimensions.