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(a) If X admits a Galois descent datum {fσ}σ∈Γ with respect to L/K, then there exists an algebraic variety Y , defined over K, and there exists a bi- σ rational isomorphism R : X → Y , defined over L, such that R = R ◦ fσ for every σ ∈ Γ. Moreover, if all the isomorphisms fσ are biregular then R can be chosen to be biregular. (b) If there is another birational isomorphism Rˆ : X → Yˆ , defined over L, σ where Yˆ is defined over K, such that Rˆ = Rˆ ◦ fσ for every σ ∈ Γ, then there exists a birational isomorphism J : Y → Yˆ , defined over K, such that Rˆ = J ◦ R.
Weil’s proof does not provide an algorithm to construct the isomorphism R explicitly. However, in the proof of [22, Proposition 1], it was observed that if {fσ} is a Galois descent datum, if each fσ is biregular and if an explicitly birational map R as before is known, then there is an explicit method to obtain a new biregular isomorphism X → Z, defined over L, with Z still defined over K. Such an explicit method is given by considering the map F : X → Y n defined by x 7→ (R(x), Rσ2 (x),...,Rσn (x)), where Γ = {σ1 = e, σ2,...,σn} and then, as there is a natural permutation action of Γ, to consider the classical invariant theory to construct a regular map Ψ : Y n → Z, defined over L, so that Z is defined over K and Ψ ◦ F : X → Z is a biregular isomorphism. In this article, we follow similar ideas as above to construct explicitly a rational map R : X → Cm, defined over L, such that Y = R(X) is defined over K and R : X → Y is a birational isomorphism. This explicit construction is done in terms of equations for X and of a Galois descent datum for X with respect to L/K. Since R is explicitly given, the algorithm can be used to compute explicit equations for Y over K. Indeed, for the sake of completeness, in the last section we will work out an example where X is a complex algebraic curve of genus five defined over Q(i). 4 This curve admits a group of conformal automorphisms isomorphic to Z2 and it is also endowed with an anticonformal involution; so, it is definable over Q.
2. Preliminaries Let K be a perfect field, let C be an algebraic closure of it, and let L be a subfield of C which is a finite Galois extension of K. We denote by Γˆ = Gal(C/K) and Γ = Gal(L/K) the Galois group associated to the extensions C/K and L/K respectively. Each η ∈ Γˆ induces a natural bijection n n ηˆ : C →C given by (y1,...,yn) 7→ (η(y1),...,η(yn)), WEIL’SGALOISDESCENTTHEOREM 3
η and if P ∈ C[z1,...,zn] then we denote by P the polynomial obtained after ap- plying η to the coefficients of P . In other words, the following diagram commutes. P Cn C ηˆ η P η Cn C Let n X = {(y1,...,yn) ∈C : Pj (y1,...,yn)=0, 1 ≤ j ≤ r} ˆ η be an affine algebraic variety where each Pj ∈ L[z1,...,zn]. If η ∈ Γ then Pj ∈ L[z1,...,zn] and n η ηˆ(X)= {(y1,...,yn) ∈C : Pj (y1,...,yn)=0, 1 ≤ j ≤ r}. Let us denote by ρ : Γˆ → Γ the canonical epimorphism defined by restriction. Note that: η σ (1) if ρ(η)= σ then Pj = Pj , and (2) if ρ(η1)= ρ(η2) thenη ˆ1(X)=η ˆ2(X). Then, if ρ(η)= σ then we denoteη ˆ(X) by Xσ.
Let {e1,...,em} be a basis of L as a K−vector space. Then the matrix
e1 e2 ··· em σ2(e1) σ2(e2) ··· σ2(em) A = . . . . ∈ M(m × m, L) . . .. . σ (e ) σ (e ) ··· σ (e ) m 1 m 2 m m is non-singular (see, for example, [13]). The trace map m Tr : L → K given by a 7→ Σj=1σj (a) extends naturally to polynomial rings
m σj Tr : L[x1,...,xn] → K[x1,...,xn] given by P 7→ Σj=1P . Lemma 1. Under the above notations, we have the following.
(a) If P ∈ L[x1,...,xn] then P ∈ SpanL(Tr(e1P ),..., Tr(emP )). σ (b) If I < L[x1,...,xn] is an ideal so that P ∈ I for every σ ∈ Γ and every P ∈ I, then I can be generated as ideal by polynomials in I ∩ K[x1,...,xn]. Proof. For each j =1,...,m we define
Qj := Tr(ejP ) ∈ L[x1,...,xn], σ and notice that Qj ∈ K[x1,...,xn] because Qj = Qj for each σ ∈ Γ. As the matrix A is non-singular, there are values λ1,...,λm ∈ L so that Aλ = E, t t where λ = [λ1 λ2 ... λm] and E = [1 0 ... 0]. In other words, we have m m Σj=1λj ej =1 and Σj=1λj σk(ej )=0, for each k =2,...,m; hence P can be written as
m m σk m λ1Q1 + ··· + λmQm = Σj=1λj ej P +Σk=2 P Σj=1λj σk(ej ) and the first statement is proved. The second statement follows from the first one σ together with the fact that, if P ∈ I for every σ ∈ Γ then Qj ∈ I. 4 RUBEN´ A. HIDALGO AND SEBASTIAN´ REYES-CAROCCA
Lemma 2. Let Y ⊂ Cn be an affine algebraic variety defined over L. If Y σ = Y for every σ ∈ Γ then Y is defined over K. More precisely, if Y is defined by the polynomials P1,...,Pr ∈ L[x1, ..., xn] then Y is also defined by the polynomials Tr(ej Pi) ∈ K[x1,...,xn], where i ∈{1,...,r} and j ∈{1,...,m}.
Proof. As Y is defined over L, its associated ideal of polynomials I < C[x1,...,xn] is generated by a finite collection of polynomials with coefficients in L. Let P ∈ I ∩ L[x1,...,xn]. Let σ ∈ Γ and let η ∈ Gal(C/K) be so that ρ(η)= σ. Then, for (b1,...,bn) ∈ Y , it holds that η σ 0= σ(P (b1,...,bn)) = η(P (b1,...,bn)) = P (η(b1),...,η(bn)) = P ◦ηˆ(b1,...,bn). As Y σ =η ˆ(Y ) and we are assuming Y σ = Y , we have thatη ˆ : Y → Y is a bijection. σ So, the above asserts that P (c1,...,cn) = 0foreach(c1,...,cn) ∈ Y . This ensures that P σ ∈ I and the desired result follows directly from Lemma 1.
3. Constructive Proof of Weil’s Galois descent theorem Let us assume that the algebraic variety X ⊂ Cn is defined by the polynomials P1,...,Pr ∈ L[x1,...,xn] and that we have a Galois descent datum
σj m {fσj : X → X }j=1 for X with respect to the Galois extension L/K, where Γ = Gal(L/K) = {σ1 = e,...,σm}. By Lemma 2, if Xσ = X for every σ ∈ Γ then X is already defined over K. Thus, from now on, we assume this is not the case.
Lemma 3. We can assume that Xσi ∩ Xσj = ∅ for i 6= j.
Proof. For each σj 6= e we may find some aj ∈ L so that σj (aj ) 6= aj . Then we may consider the algebraic variety Xˆ ⊂Cn+m−1 defined by the polynomials
P1(x1,...,xn)= ··· = Pr(x1,...,xn)=0, xn+1 = a2,...,xn+m−1 = am. Clearly, the map
Q : X → Xˆ given by (x1,...,xn) 7→ (x1,...,xn,a2,...,am) defines a biregular isomorphism whose inverse is provided by the projection −1 Q : Xˆ → X given by (x1,...,xn,a2,...,am) → (x1,...,xn). By the construction, we see that Xˆ σi ∩Xˆ σj = ∅ for every i 6= j. It is easyto check σ −1 that a Galois descent datum for Xˆ is given by {gσ = Q ◦ fσ ◦ Q : σ ∈ Γ}. 3.1. A first isomorphism. Let us consider the rational map n Φ: X → Πσ∈ΓC defined by x 7→ (fσ(x))σ∈Γ . If f (x) = ( r1,σ (x) ,..., rn,σ(x) ), where s and r are relatively prime polyno- σ s1,σ (x) sn,σ(x) j,σ j,σ mials (each one defined over L), then the equality
fσ(ye)= yσ = (y1,σ,...,yn,σ) provides n polynomial equations
yj,σsj,σ(ye)= rj,σ(ye). WEIL’SGALOISDESCENTTHEOREM 5
We see that Φ(X) is the affine algebraic variety defined over L given by σ σ Φ(X)= {(yσ)σ∈Γ : P1 (yσ)= ··· = Pr (yσ)=0, yj,σsj,σ(ye)= rj,σ(ye), σ ∈ Γ, j =1,...,n}. Lemma 4. The map Φ: X → Φ(X) is a birational isomorphism. Furthermore, Φ is biregular provided that each fσ is polynomial.
Proof. As each fσ is a birational isomorphism, Φ induces a birational isomorphism between X and Φ(X). The inverse map is given by projection in the first coordinate (so it is regular) n n π : Πσ∈ΓC →C given by (yσ)σ∈Γ 7→ ye.
If each fσ is a polynomial, then Φ is a regular map and the lemma follows.
n 3.2. A linear permutation action on Πσ∈ΓC induced by Γ. Let us consider the following natural permutation action of Γ on the σ−coordinates: n n Θ:Γ × Πσ∈ΓC → Πσ∈ΓC given by τ, (yσ)σ∈Γ 7→ (yτσ)σ∈Γ Lemma 5. If τ 6= e then Θ(τ)(Φ(X)) ∩ Φ(X)= ∅.
Proof. Let (yσ)σ∈Γ ∈ Θ(τ)(Φ(X)) ∩ Φ(X). The condition (yσ)σ∈Γ ∈ Φ(X) ensures −1 σ τ that yσ ∈ X and, in particular, yτ −1 ∈ X . Moreover, the condition (yσ)σ∈Γ ∈ −1 Θ(τ)(Φ(X)) ensures that Θ(τ )(yσ)σ∈Γ = (yτ −1σ)σ∈Γ ∈ Φ(X), that is, yτ −1 ∈ X. −1 The above implies that X ∩ Xτ 6= ∅, which is not possible by Lemma 3.
3.3. A linearization of the Galois descent datum. Let τ ∈ Γ and let η ∈ τ Gal(C/K) be so that ρ(η)= τ. As fτσ = fσ ◦ fτ , we see that τ τ (Φ ◦ fτ ) (x) = (fσ (fτ (x)))σ∈Γ = (fτσ(x))σ∈Γ = (Θ(τ) ◦ Φ) (x) τ η −1 n and, as fσ = fσ =η ˆ ◦ fσ ◦ ηˆ , whereη ˆ acts on Πσ∈ΓC , we obtain the following equality −1 −1 −1 τ Φ ◦ ηˆ ◦ fτ (x)) = fσ(ˆη (fτ (x))) σ∈Γ = ηˆ (fσ (fτ (x))) σ∈Γ = −1 −1 = ηˆ (fτσ(x)) σ∈Γ = ηˆ ◦ Θ(τ) ◦ Φ (x). The above can be summarized in the following commutative diagram. Φ X Φ(X)
fτ Θ(τ) Φτ Xτ Θ(τ)(Φ(X))=Φτ (Xτ )=Φ(X)τ ηˆ−1 ηˆ−1 Φ Φ(X) X (3.1) Similarly, it is not difficult to see that, for every τ ∈ Γ and every η ∈ Gal(C/K), we have that Θ(τ) ◦ ηˆ =η ˆ ◦ Θ(τ). 6 RUBEN´ A. HIDALGO AND SEBASTIAN´ REYES-CAROCCA
Remark 1. Note that, from the commutative diagram (3.1), each isomorphism fσ : X → Xσ is transformed into the linear (permutation) isomorphism Θ(σ):Φ(X) → σ Φ(X) . So, the Galois descent datum {fσ}σ∈Γ for X is now transformed into the (linear) Galois descent datum {Θ(σ)}σ∈Γ. In other words, the above method says that we are able to change our model of X (in an explicit manner) to W = Φ(X) for which the Galois descent datum is given by permutation linear transformations. 3.4. A second isomorphism: Invariant theory. As Θ(Γ) =∼ Γ is a finite group of permutations, it follows from Hilbert-Noether’s theorem (see [16, 17] and also n Θ(Γ) [19, Ch. 14]) that the algebra C[Πσ∈ΓC ] of Θ(Γ)-invariant polynomials with coefficients in C, is finitely generated. Let us consider a finite set of generators of such an C-algebra, say Θ(Γ) E1((yσ)σ∈Γ),...,EN ((yσ)σ∈Γ) ∈C[(yσ)σ∈Γ] . At this point it is important to note that the finite permutation of coordinates produced by the action of Θ(Γ) does not depend on the field C, that is, we may consider this permutation action on the product space Πσ∈ΓB, where B is the basic field of C. It follows, in particular, that
E1((yσ)σ∈Γ),...,EN ((yσ)σ∈Γ) ∈ K[(yσ )σ∈Γ]. For the constructiveness part we need to have explicitly computed such polyno- mials Ej (this can be done, for instance, with MAGMA [2] or Macauley2 [7]). Consider the regular map n N Ψ : Πσ∈ΓC →C given by (yσ)σ∈Γ 7→ (E1((yσ)σ∈Γ),...,EN ((yσ)σ∈Γ)). Lemma 6. The regular map Ψ satisfies the following properties: (a) Ψσ =Ψ for every σ ∈ Γ. (b) Ψ ◦ Θ(σ)=Ψ for every σ ∈ Γ. (c) if Ψ(w)=Ψ(z) then there is some σ ∈ Γ so that w = Θ(σ)(z). n N (d) Y = Ψ(Φ(X)) and Ψ(Πσ∈ΓC ) are algebraic subvarieties of C . Proof. Properties (a) and (b) are easy to check. Property (c) is consequence of the fact that (i) a finite group is a reductive group [11, 18] and (ii) for a reductive group G, say acting linearly over Cd, and a set of generators of the G−invariant d m polynomials, say I1,...,Im ∈ C[z1,...,zd], the map A = (I1,...,Im): C → C turns out to be a regular branched cover map with G as its deck group. In other words, A(p) = A(q) if and only if there exists some α ∈ G so that α(p) = q; the branch values of A agree with the a−images of those points with non-trivial stabilizer (for details see, for instance, [11, 18]). Property (d) follows from the previous ones. n Remark 2. The algebra of regular maps on Ψ(Πσ∈ΓC ) is known to be isomorphic Θ(Γ) to the algebra C[(yσ)σ∈Γ] of symmetric polynomials with respect to the linear group Θ(Γ); see [4]. This can be seen by considering the surjective homomorphism Θ(Γ) ξ : C[t1,...,tN ] →C[(yσ)σ∈Γ] defined by ξ(p)= p(E1,...,EN ). Properties (b) and (c) of Lemma 6 assert that Ψ is a finite regular (branched) cover with the finite algebraic group Θ(Γ) as its deck group. Lemma 7. The map Ψ : Φ(X) → Y is a biregular isomorphism. In particular, R =Ψ ◦ Φ: X → Y is a birational isomorphism; it is biregular provided that each fσ is polynomial. WEIL’SGALOISDESCENTTHEOREM 7
Proof. Set Φ(X)= W . Since for each τ ∈ Γ −{e} one has that Θ(τ)(W ) ∩ W = ∅ (see Lemma 5), the polynomial map Ψ : W → Y is bijective. The set
Wˆ := ∪σ∈ΓΘ(σ)(W ) is a reducible affine variety whose (pairwise disjoint) irreducible components are Θ(σ)(W ), for σ ∈ Γ. Since these irreducible components are pairwise disjoint, we may see that the algebra of regular maps on Wˆ , say C[Wˆ ], is the product of the algebras of regular maps of the components, that is,
C[Wˆ ] = Πσ∈ΓC[Θ(σ)(W )].
The above isomorphism is given by the restriction of each regular map of Wˆ to each of its irreducible components. Note that there is natural isomorphism
ρ : C[W ] →C[Wˆ ]Θ(Γ), where C[Wˆ ]Θ(Γ) denotes the sub-algebra of Θ(Γ)-invariant regular maps on Wˆ . This isomorphism is given as follows. If p ∈C[W ] then for each σ ∈ Γ we may consider −1 the regular map ρσ(p)= p ◦ Θ(σ) ∈C[Θ(σ)(W )]. Then ρ(p) := (ρσ(p))σ∈Γ turns out to be an injective homomorphism. It is clear that every Θ(Γ)−invariant regular map of Wˆ is obtained in that way (so ρ is surjective). On the other hand, C[Y ] is isomorphic to C[Wˆ ]Θ(Γ). To see this, one may consider the injective homomorphism χ : C[Y ] →C[Wˆ ]Θ(Γ) defined by χ(p)= p ◦ Ψ. Now, to see that χ is onto we only need to note that ρ−1(χ(C[Y ])) is a sub-algebra of C[W ], that W is irreducible and that Y has the same dimension as W . In this way, χ−1 ◦ ρ produces an isomorphism between C[W ] and C[Y ], that is, Ψ :Φ(X) → Y is a biregular isomorphism and, in particular, R =Ψ ◦ Φ: X → Y is a birational isomorphism. As Ψ : W → Y is biregular isomorphism and Φ : X → W is biregular if each fσ is polynomial, then R : X → Y turns out to be biregular provided that each fσ is polynomial.
Lemma 8. Y is defined over K.
Proof. Let τ ∈ Γ and let η ∈ Gal(C/K) be so that ρ(η) = τ. It follows from part (a) of Lemma 6 that Ψ ◦ ηˆ =η ˆ ◦ Ψ. n n Now one sees that the bijectionη ˆ : Πσ∈ΓC → Πσ∈ΓC descends to the bijection ηˆ : CN →CN . Then it follows from diagram (3.1) thatη ˆ(Y )= Y , that is, Y τ = Y . As this holds for every τ ∈ Γ, it follows from Lemma 2 that Y is defined over K.
Remark 3. Above we have constructed an explicit birational isomorphism R : X → Y . We should be able to construct explicit equations for Y , say given by poly- nomials Q1,...,Qm ∈ L[t1,...,tN ] (again, this can be done by using MAGMA). As we already know that Y is defined over K, so Y is also defined by some polyno- mials over K. To obtain these polynomials over K, we proceed to replace each of the polynomials Qj, which is not already defined over K, by the set of polynomi- als Tr(e1Qj),..., Tr(emQj ) ∈ K[x1,...,xn], where {e1,...,em} is a basis of L as K−vector space (see the proof of Lemma 1). 8 RUBEN´ A. HIDALGO AND SEBASTIAN´ REYES-CAROCCA
4. Applications to complex algebraic varieties and field of moduli Weil’s Galois descent theorem has also been used in the study of complex alge- braic varieties and their fields of moduli; in particular, in the case of Belyi curves and dessin d’enfants (see, for example [23]). Let us denote by Gal(C) the group of field automorphisms of the field of complex numbers and let X be a complex algebraic variety with a finite group Aut(X) of automorphisms. Let ΓX be the subgroup of Gal(C) consisting of those elements σ with the property that Xσ and X are C-isomorphic. The field of moduli of X, denoted by M(X), is the fixed field of ΓX . In general, the determination of whether the field of moduli is a field of definition is a difficult task, even in the case of algebraic curves; see, for example [1, 5, 9, 10, 12, 20, 21, 23]. σ A Galois descent datum for X is a collection of isomorphisms fσ : X → X η defined over C such that for every pair η, τ ∈ ΓX it holds that fητ = fτ ◦ fη. We shall suppose that X admits a Galois descent datum and that X is defined over a finite Galois extension L of M(X) (we remark that the second assumption is vacuous in the case of complex algebraic curves; see [8] and [14]). Let L¯ be the algebraic closure of L in C (which is also algebraic closure of M(X)). As σ ∈ ΓX acts as the identity on M(X), it defines an automorphism of L¯. Galois theory implies that σ ∈ ΓX acts a field automorphism of L. Now, as Aut(X) is finite, it can be seen that every automorphism of X is defined over L¯. If τ ∈ Aut(C/L¯) then Xτ = X and, in particular, τ ∈ ΓX and fτ ∈ Aut(X). τ σ Let σ ∈ ΓX . If τ ∈ Aut(C/L) then fσ : X → X . It follows that there is some τ ¯ hτ ∈ Aut(X) so that fσ = fσ ◦ hτ . This ensures that fσ is defined over L. Let us consider the map ¯ −1 Θ : Aut(C/L) → Aut(X) given by σ 7→ fσ .
Since, for each σ ∈ ΓX the isomorphism fσ is defined over L¯, the condition η fητ = fτ ◦ fη ensures that the above is a homomorphism of groups. The kernel of Θ is a finite index subgroup of Aut(C/L¯) and its fixed field is then a finite extension of L¯; so equals to L¯. This implies that the kernel is the whole group Aut(C/L¯). In particular, if τ, σ ∈ ΓX have the same restriction to L¯ then fτ = fσ. Now, since L is a finite Galois extension of M(X), we may see that there are only a finite number of algebraic varieties of the form Xτ which are isomorphic to X, where τ ∈ Gal(C). In addition, we have a finite number of possible isomorphisms fσ for σ ∈ ΓX . Thereby, we might assume that X and all these isomorphisms are defined over L. In this way, Weil’s Galois descent theorem asserts the following.
Corollary 1. Let X be a complex algebraic variety. Assume that Aut(X) is finite and that X is defined over a finite extension of its field of moduli M(X). Then
(a) X admits a Galois descent datum, say {fσ}σ∈ΓX , if and only if there is an algebraic variety Y defined over M(X) and a birational isomorphism R : σ X → Y defined over the algebraic closure of M(X), such that R = R ◦ fσ for every σ ∈ ΓX . If, moreover, all the isomorphisms fσ are biregular, then R can be chosen to be biregular. (b) If there is another isomorphism Rˆ : X → Yˆ , where Yˆ is defined over σ M(X) and such that Rˆ = Rˆ ◦ fσ, for every σ ∈ ΓX then there exists an isomorphism J : Y → Yˆ defined over M(X) so that Rˆ = J ◦ R. WEIL’SGALOISDESCENTTHEOREM 9
5. An Explicit Example: A curve of genus 5 ∼ 4 A closed Riemann surface S of genus 5 admitting a group H = Z2 of conformal automorphisms is called a classical Humbert’s curve. We may identify S/H with the Riemann sphere together with five cone points, which, up to conjugation by a M¨obius transformation, can be supposed to be {∞, 0, 1, λ1, λ2}. In [3] it was proved that S can be represented by the following irreducible and non-singular complex projective curve 2 2 2 u1 + u2 + u3 = 0 2 2 2 4 λ1u1 + u2 + u4 = 0 ⊂ PC. 2 2 2 λ2u1 + u2 + u5 = 0
We may consider an affine model by taking u1 = 1. For our example we consider λ1 = −1 and λ2 = i. If we set u2 = x1, u3 = x2, u4 = x3 and u5 = x4, then the affine model (defined over Q(i)) is given by 2 2 1 + x1 + x2 = 0 2 2 4 X : −1 + x1 + x3 = 0 ⊂ C . 2 2 i + x1 + x4 = 0 Notice that the involution 4 4 L : C → C given by (x1, x2, x3, x4) 7→ (−i x1, −i x3, −i x2, −i x4) keeps invariant X, so it defines a symmetry of X. It follows from Weil’s Galois descent theorem that X is in fact definable over K = Q = Q(i) ∩ R. Let us consider Gal(Q(i)/Q)= {σ1(z)= z, σ2(z)= z} and set
σ2 fσ2 : X → X = X given by (x1, x2, x3, x4) 7→ (ix1, ix3, ix2, ix4). In this example, 4 8 Φ: X ⊂ C → Φ(X) ⊂ C : (x1, x2, x3, x4) 7→ (x1, x2, x3, x4, ix1, ix3, ix2, ix4). Therefore, the equations defining Φ(X) are given by
z1 − ix1 =0, z2 − ix3 =0 z3 − ix2 =0, z4 − ix4 =0 2 2 2 2 2 2 . 1+ x1 + x2 =0, −1+ x1 + x3 =0, i + x1 + x4 =0
We have that R : X → Y = R(X) : (x1, x2, x3, x4) 7→ (t1,...,t14), where 2 t1 = (1+ i)x1, t2 = x2 + ix3, t3 = x3 + ix2, t4 = (1+ i)x4, t5 = ix1, 2 t6 = x2 + ix3, t7 = ix2x3, t8 = ix4, t9 = x1x2 − x1x3, t10 = x1x3 − x1x2, t11 =0, t12 =0, t13 = x2x4 − x3x4, t14 = x3x4 − x2x4. The inverse map R−1 : Y → X is given as t t − it t − it t (t ,...,t ) 7→ (x , x , x , x )= 1 , 2 3 , 3 2 , 4 . 1 14 1 2 3 4 1+ i 2 2 1+ i Explicit equations for Y are given by 2 2 4+ t2 − t3 =0, 2 t1 + t2t3 =0, 2 2 t1 + t4 − 2=0, Y := t14 = −t13 = t4(t3 − t2)/2, t12 = t11 =0, 2 t10 = −t9 = −t1(t2 − t3)/2, t8 = t4/2, 2 2 2 t7 = (t2 + t3)/4, t6 = t2, t5 = t1/2. 10 RUBEN´ A. HIDALGO AND SEBASTIAN´ REYES-CAROCCA
The above, in particular, asserts that (by forgetting the coordinates tj , for j ≥ 5) that the algebraic curve 2 2 4+ w2 − w3 =0 ˆ 2 4 Y = w1 + w2w3 =0 ⊂ C 2 2 w1 + w4 − 2=0 is isomorphic to X by the isomorphism
Rˆ : X → Yˆ given by (x1, x2, x3, x4) 7→ (t1,t2,t3,t4) = (w1, w2, w3, w4). Remark 4. In this example, the MAGMA routine we can use is the following. > Q:=Rationals( ); > P
t14 = −t13 = t4(t3 − t2)/2, t12 = t11 =0 t9 + t10 =0, t6 − t7 =0 t5 + t8 − 1=0, 2 2 2 t4 − 2t8 =0, t3 − 2t7 − 2=0, t2 − 2t7 +2=0, 2 2 2 2 1 4 2 t8t10 + t8 − 2t8t10 − 2t8 − 4 t10 + t10 +1=0, 2 2 2 t7t10 + t8t10 +2t8 − t10 − 2=0, 2 1 2 t7t8 − t7 − t8 +2t8 + 2 t10 − 1=0, . 2 2 t7 − t8 +2t8 − 2=0, t2t7 + t2 − t3t8 + t3 =0, 1 t2t3 − 2t8 +2=0, t1 − 1/2t2t10 − 2 t3t10 =0, 2 2 t2t10 − 2t3t7 +2t3t8 + t3t10 =0, t2t8 − t2 − t3t7 + t3 =0. References
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Departamento de Matematicas´ y Estad´ıstica, Universidad de La Frontera, Avenida Francisco Salazar 01145, Temuco, Chile. E-mail address: [email protected], [email protected]