Generalized Fermat Curves

Home , Fermat curve

ARTICLE IN PRESS JID:YJABR AID:12369 /FLA [m1G; v 1.77; Prn:15/01/2009; 8:13] P.1 (1-18) JournalofAlgebra••• (••••) •••–•••

1 Contents lists available at ScienceDirect 1 2 2 3 3 4 Journal of Algebra 4 5 5 6 6 7 www.elsevier.com/locate/jalgebra 7 8 8 9 9 10 10 11 11 12 Generalized Fermat curves 12 13 13 a,∗,1 b,2 c 14 Gabino González-Diez ,RubénA.Hidalgo , Maximiliano Leyton 14 15 15 a Departamento de Matemáticas, UAM, Madrid, Spain 16 16 b Departamento de Matemáticas, Universidad Técnica Federico Santa María, Valparaíso, Chile 17 c Institut Fourier-UMR5582, Université Joseph Fourier, France 17 18 18 19 19 article info abstract 20 20 21 21 Article history: 22 A closed Riemann surface S is a generalized Fermat curve of type 22 Received 12 November 2007 k n if it admits a group of automorphisms H =∼ Zn such that the 23 ( , ) k 23 Available online xxxx quotient O = S/H is an orbifold with signature (0,n + 1;k,...,k), 24 Communicated by Corrado de Concini 24 that is, the Riemann sphere with (n + 1) conical points, all of same 25 25 order k.ThegroupH is called a generalized Fermat group of type 26 Keywords: 26 Riemann surfaces (k,n) and the pair (S, H) is called a generalized Fermat pair of 27 27 Fuchsian groups type (k,n). We study some of the properties of generalized Fermat 28 Automorphisms curves and, in particular, we provide simple algebraic curve real- 28 29 ization of a generalized Fermat pair (S, H) in a lower-dimensional 29 30 projective space than the usual canonical curve of S so that the 30 31 normalizer of H in Aut(S) is still linear. We (partially) study the 31 32 problem of the uniqueness of a generalized Fermat group on a 32 33 fixedRiemannsurface.Itisnotedthatthemodulispaceofgeneral- 33 ized Fermat curves of type (p,n),wherep is a prime, is isomorphic 34 34 to the moduli space of orbifolds of signature (0,n + 1; p,...,p). 35 35 Some applications are: (i) an example of a pencil consisting of only 36 3 2 36 non-hyperelliptic Riemann surfaces of genus gk = 1 + k − 2k ,for 37 every integer k 3, admitting exactly three singular fibers, (ii) an 37 38 38 injective holomorphic map ψ : C −{0, 1}→Mg ,whereMg is the 39 moduli space of genus g 2 (for infinitely many values of g), and 39 40 (iii) a description of all complex surfaces isogenous to a product 40 = = Z2 Z4 41 with invariants p g q 0 and covering group equal to 5 or 2. 41 42 © 2009 Published by Elsevier Inc. 42 43 43 44 44 45 45 46 46 47 47 48 * Corresponding author. UNCORRECTED PROOF 48 49 E-mail addresses: [email protected] (G. González-Diez), [email protected] (R.A. Hidalgo), 49 50 [email protected] (M. Leyton). 50 1 51 Partially supported by project BFM2003-04964 of the MCYT. 51 2 52 Partially supported by projects Fondecyt 1070271 and UTFSM 12.08.01. 52 53 53 0021-8693/$ – see front matter © 2009 Published by Elsevier Inc. 54 doi:10.1016/j.jalgebra.2009.01.002 54

1 1. Preliminaries 1 2 2 3 If S denotes a closed Riemann surface, then Aut(S) will denote its full group of conformal auto- 3 4 4 morphisms. If H < Aut(S),thenwedenotebyAutH (S) the normalizer of H inside Aut(S),thatis,the 5 highest subgroup of Aut(S) containing H as normal subgroup. If K is any group, we denote by K its 5 6 commutator subgroup. 6 7 7 An orbifold O of signature (g, r;n1,...,nr ), where g 0, r 0, n j 2, are integers, is a closed 8 8 Riemann surface of genus g together with a collection of r conical points of orders n1,...,nr .Its 9 orb O 9 orbifold fundamental group, π1 ( ), is a group with generators a1,...,ag , b1,...,bg , c1,...,cr and 10 g [ ] r = = n1 =···= nr [ ]= −1 −1 10 relations j=1 a j, b j s=1 c j 1 c1 cr , where a, b aba b . Generalities on orbifolds 11 11 can be found in [26,29]. 12 12 An orbifold of signature (g, 0;−) is a closed Riemann surface of genus g. A conformal automor- 13 13 phism of an orbifold is a conformal automorphism of the corresponding Riemann surface which 14 14 preserves the conical points of the same orders. The homology cover of an orbifold O is an orb- 15 15 ifold O providing the highest regular Abelian cover of it, that is, the regular covering induced by the 16 16 commutator subgroup π orb(O) ¡ π orb(O). In many cases O has no conical points, that is, it turns 17 1 1 17 out to be a closed Riemann surface. 18 18 A closed Riemann surface S is called a generalized Fermat curve of type (k,n) if it is the homology 19 19 cover of an orbifold of signature (0,n + 1;k,...,k). It follows that there exists H < Aut(S), so that 20 20 H =∼ Zn and S/H is an orbifold with signature (0,n + 1;k,...,k). In this case we say that H is a 21 k 21 generalized Fermat group of type (k,n) and (S, H) a generalized Fermat pair of type (k,n). Reciprocally, 22 22 each pair (S, H) so that S is a closed Riemann surface, H < Aut(S) is isomorphic to Zn and S/H is 23 k 23 an orbifold with signature (0,n + 1;k,...,k) is a generalized Fermat pair of type (k,n). Riemann– 24 24 Hurwitz’s formula asserts that the genus gk,n of a generalized Fermat curve of type (k,n) is 25 25 26 26 n−1 27 2 + k ((n − 1)(k − 1) − 2) 27 gk,n = . (1) 28 2 28 29 29 30 In particular, a generalized Fermat curve of type (k,n) is hyperbolic, that is, it has the hyperbolic 30 31 plane H2 as universal cover Riemann surface, if and only if (n − 1)(k − 1)>2. As an example, the 31 32 classical Fermat curve xk + yk + zk = 0 defines, in the complex projective plane P2,ageneralized 32 33 Fermat curve of type (k, 2). Examples of generalized Fermat curves of type (2,n) were studied in [9] 33 34 from the point of view of Fuchsian and Schottky groups (see also Section 6). 34 35 We say that two generalized Fermat pairs (S1, H1) and (S2, H2) are topologically (holomorphically) 35 36 equivalent if there is some orientation-preserving homeomorphism (holomorphic homeomorphism) 36 −1 37 f : S1 → S2 so that fH1 f = H2. 37 38 The non-hyperbolic generalized Fermat pairs are the following ones. 38 39 39 40 (i) (k,n) = (2, 2): S = C and H =A(z) =−z, B(z) = 1/z. 40 41 = = C = = + = + 2π i/3 41 (ii) (k,n) (3, 2): S /Λe2πi/3 , where Λe2πi/3 A(z) z 1, B(z) z e , and H is generated 42 by the induced transformations of J(z) = e2π i/3z and T (z) = z + (2 + e2π i/3)/3. In this case, the 3 42 43 cyclic groups J, TJ and T 2 J project to the only 3 cyclic subgroups in H with fixed points; 43 44 3 fixed points each one. This is provided by the degree 3 Fermat curve x3 + y3 + z3 = 0. 44 45 (iii) (k,n) = (2, 3): S = C/Λτ , where τ ∈ C with Im(τ )>0, Λτ =A(z) = z + 1, B(z) = z + τ , and 45 46 H is generated by the induced transformations from T1(z) =−z, T2(z) =−z + 1/2andT3(z) = 46 47 −z + τ /2. In this case, the conformal involutions induced on the torus by T1, T2, T3 and their 47 48 product are the only ones acting with fixed points; 4 fixed points each. This is also described by 48 49 the algebraic curveUNCORRECTED{x2 + y2 + z2 = 0,λx2 + y2 + w2 = 0}, PROOF where λ ∈ C −{0, 1}. 49 50 50 51 The hyperbolic generalized Fermat pair (S, H) of type (k,n) can be described in terms of Fuchsian 51 52 groups as follows. Classical uniformization theorem asserts that the orbifold O = S/H is uniformized 52 53 by a Fuchsian group Γ

Please cite this article in press as: G. González-Diez et al., Generalized Fermat curves, J. Algebra (2009), doi:10.1016/j.jalgebra.2009.01.002 ARTICLE IN PRESS JID:YJABR AID:12369 /FLA [m1G; v 1.77; Prn:15/01/2009; 8:13] P.3 (1-18) G. González-Diez et al. / Journal of Algebra ••• (••••) •••–••• 3 1 = k =···= k = ··· = 1 Γ x1,...,xn+1: x1 xn+1 x1x2 xn+1 1 . (2) 2 2 3 There is a torsion free normal subgroup L ¡ Γ providing a uniformization of S and so that H = 3 4 Γ/L. The commutator subgroup Γ is torsion free [22] and uniformizes a closed Riemann surface S . 4 5 On S we have the Abelian group H = Γ/Γ =∼ Zn so that S /H = O, that is, S is a generalized 5 k 6 Fermat curve of type (k,n) with generalized Fermat group H of type (k,n).AsL ¡ Γ satisfies that 6 7 Γ/L is Abelian, we must have that Γ ¡ L.AstheindexofL in Γ is equal to the index of Γ in Γ ,we 7 8 have that L = Γ .AsFuchsiangroupsofafixedtype(k,n) are topologically rigid and the commutator 8 9 subgroup Γ is a characteristic subgroup, all the above can be summarized in the following. 9 10 10 11 11 Theorem 1. Let (S, H) be a generalized Fermat pair and Γ be a (orbifold) universal cover group of the orb- 12 12 ifold S/H. Then (S, H) is holomorphically equivalent to (U /Γ, Γ /Γ ),whereU∈{C, C, H2} is the universal 13 13 Riemann surface cover of S/HandΓ is the commutator subgroup of Γ . In particular, any two generalized 14 14 Fermat pairs of the same type are topologically equivalent. 15 15 16 16 Denoting by a the congruence class of x mod Γ , we easily obtain the following consequences of 17 j j 17 Theorem 1, which we will need later. 18 18 19 19 : → 20 Corollary 2. Let (S, H) be a generalized Fermat pair of type (k,n) and let P S S/H be a branched regular 20 21 covering with H as group of cover transformations. 21 22 22 23 1.- If AutH (S) denotes the normalizer of H inside Aut(S), then each orbifold automorphism of S/Hliftstoan 23 : → 24 automorphism in AutH (S); that is, for each orbifold automorphism τ S/H S/H there is a conformal 24 : → = 25 automorphism τ S SsothatPτ τ P. 25 26 2.- There exist elements of order k in H, say a1,...,an,sothat: 26 = 27 (i) H a1,...,an ; 27 = ··· n−1 28 (ii) each a1,...,an and an+1 a1a2 an has exactly k fixed points; 28 ∈ ∈ ∪···∪ ∪ 29 (iii) if h H has fixed points, then h a1 an an+1 ; 29 ∈ 30 (iv) if k is prime and h H has no fixed points and it has order k, then no non-trivial power of h has fixed 30 31 points; and 31 ∈ 32 (v) if h H is an element of order k with fixed points and x, y are any two of these fixed points, then 32 ∗ ∈ ∗ = 33 there is some h Hsothath (x) y. 33 34 Such a set of generators a1,...,an shall be called a standard set of generators for the generalized Fermat 34 35 group H. 35 36 36 37 Remark 3. If (S, H) is a generalized Fermat pair of type (k,n),thenforeachh ∈ H of order k with 37 38 fixed points, we have that the Riemann surface structure of the orbifold R = S/h is a generalized 38 39 Fermat curve of type (k,n − 1) with K = H/h as a generalized Fermat group of type (k,n − 1).This 39 40 fact permits to construct towers of generalized Fermat curves starting from some non-hyperbolic one 40 41 and adding an extra conical point at each step. 41 42 42 43 Having recalled all the necessary general basic facts and descriptions, we will proceed in the rest 43 44 of this paper as follows. In Section 2 we note that generalized Fermat curves of genus at least 2 are 44 45 non-hyperelliptic and we provide algebraic curve description of them (as fiber products of classical 45 46 Fermat curves) in lower-dimensional projective spaces than the usual canonical curve (this may be 46 47 of interest for computation on Riemann surfaces). In Section 3 we discuss the problem of uniqueness 47 48 of generalized Fermat groups on each generalized Fermat curve. In Section 4 we describe the locus 48 49 of generalized Fermat curvesUNCORRECTED in the corresponding moduli space. PROOF In Section 5 we proceed to use the 49 50 results obtained in the previous sections in order to produce three examples. The first one is the 50 3 2 51 construction of a pencil of non-hyperelliptic Riemann surfaces of genus gk = 1 + k − 2k ,forevery 51 52 integer k 3, with exactly three singular fibers. The second one is the construction of an injective 52 53 holomorphic map ψ : C −{0, 1}→Mg , where Mg is the moduli space of genus g 2, for infinitely 53 54 many values of g. The third one is the description of all complex surfaces isogenous to a product 54

1 = × = = = Z2 = Z4 1 X S1 S2/G with invariants p g q 0 and group G equals either G 5 or G 2. Finally, 2 in Section 6 we describe those generalized Fermat curves which can act as group of isometries of 2 3 hyperbolic handlebodies. 3 4 4 5 2. Algebraic description 5 6 6 7 2.1. Non-hyperellipticity of general Fermat curves 7 8 8 9 Let us recall that a closed Riemann surface S of genus g 2 is called hyperelliptic if it admits a 9 10 (necessarily unique) conformal involution with exactly 2(g + 1) fixed points, called the hyperellip- 10 11 tic involution.Equivalently,S is hyperelliptic if and only if there is a two-fold branched regular 11 12 : → C 12 covering f S . The algebraic equation of a hyperelliptic Riemann surface S is of the form 13 2 2(g+1) 13 y = = (x − a j) and, moreover, as the hyperelliptic involution belongs to the center of Aut(S), 14 j 1 14 a complete description of Aut(S) can be done via such a curve description [6]. 15 15 Unfortunately, generalized Fermat curves of genus at least 2 are non-hyperelliptic Riemann sur- 16 16 faces. 17 17 18 18 19 Theorem 4. A hyperbolic generalized Fermat curve is non-hyperelliptic. 19 20 20 21 Proof. Let (S, H) be a hyperbolic generalized Fermat pair of type (k,n). Let us assume S is hyperellip- 21 22 tic and let us denote by j its hyperelliptic involution. As j belongs to the center of Aut(S), the action 22 23 of H descends to a finite Abelian group of Möbius transformations PSL(2, C) on the Riemann sphere. 23 24 The fact that the finite Abelian groups inside PSL(2, C) are either finite cyclic groups or the Klein 24 25 Z2 = = = 2 25 group 2 obligates to have either (i) k > 2 and n 1, or (ii) k 2 and n 1, 2, a contradiction. 26 26 27 2.2. Algebraic description 27 28 28 29 As a generalized Fermat curve S of genus g 2 is non-hyperelliptic, it is well known that, if 29 30 1,0 30 w1,...,w g is any basis of H (S),theg-dimensional vector space of holomorphic 1-forms, then 31 g−1 31 θ : S → P defined by θ(x) =[w1(x) :···: w g (x)], is a holomorphic embedding and θ(S) is a non- 32 singular projective algebraic curve of degree 2(g − 1), called a canonical curve of S (see, for instance, 32 33 [10]). 33 34 34 Assume (S, H) is generalized Fermat pair of type (k,n) and genus g = gk,n 2. Theorem 5 below 35 provides a holomorphic embedding ρ : S → Pn so that ρ(S) is a non-singular projective algebraic 35 36 n−1 36 curve of degree k 2(g − 1) and the action of H is defined by a projective linear action H0 < 37 37 PGL(n + 1, C). Note that this description produces a projective curve of lower degree than a canonical 38 38 curve and also the embedding is in lower dimension than the canonical embedding (this lower degree 39 39 and dimension representation seems to be useful for practical computations). Not only the action of 40 40 H will be linear, but also the action of AutH (S). 41 41 Let us start with the following fiber product of (n − 1) classical Fermat curves, say the algebraic 42 n 42 curve C(λ1,...,λn−2) ⊂ P given by the following (n − 1) homogeneous polynomials of degree k 43 43 44 ⎧ 44 ⎪ xk + xk + xk = 0, 45 ⎪ 1 2 3 45 ⎪ 46 ⎪ λ xk + xk + xk = 0, 46 ⎨⎪ 1 1 2 4 47 k k k 47 λ2x + x + x = 0, 48 ⎪ 1 2 5 (3) 48 UNCORRECTED⎪ . . . PROOF 49 ⎪ . . . 49 ⎪ . . . 50 ⎩⎪ 50 k + k + k = 51 λn−2x1 x2 xn+1 0, 51 52 52

53 where λ j ∈ C −{0, 1}, λi = λ j ,fori = j. The conditions on the parameters λ j ensure that 53 54 C(λ1,...,λn−2) is a non-singular projective algebraic curve, that is, a closed Riemann surface. On 54

∼ 1 − = Zn 1 C(λ1,...,λn 2) we have the Abelian group H0 k of conformal automorphisms generated by the 2 transformations 2 3 3 4 4 [ :···: + ] =[ :···: − : : + :···: + ] = 5 a j x1 xn 1 x1 x j 1 ωkx j x j 1 xn 1 , j 1,...,n, 5 6 6 2π i/k n 7 where ωk = e . If we consider the degree k holomorphic map 7 8 8 9 9 π : C(λ1,...,λn−2) → C 10 10 11 11 12 given by 12 13 13 k 14 14 [ :···: ] =− x2 15 π x1 xn+1 , 15 x1 16 16 17 17 then π ◦a j = π ,foreverya j , j = 1,...,n. It follows that C(λ1,...,λn−2) is a generalized Fermat curve 18 18 with generalized Fermat group H0 with standard generators a1,...,an, an+1 = a1a2 ···an.Thefixed 19 19 points of a j on C(λ1,...,λn−2) are given by the intersection Fix(a j) = F j ∩ C(λ1,...,λn−2) where 20 20 F j ={x j = 0}. In this way, the branch values of π aregivenbythepoints 21 21 22 22 23 π Fix(a1) =∞, π Fix(a2) = 0, π Fix(a3) = 1, 23 24 24 25 25 26 26 π Fix(a ) = λ , ..., π Fix(a + ) = λ − . 27 4 1 n 1 n 2 27 28 28 29 As every generalized Fermat pair (S, H) is uniquely determined by the orbifold S/H (by Theo- 29 30 rem 1), we get the following algebraic description of generalized Fermat pairs. 30 31 31 32 Theorem 5. Let (S, H) be a generalized Fermat pair of type (k,n) and, up to a Möbius transformation, let 32 33 33 34 34 {∞, 0, 1,λ1,λ2,...,λn−2} 35 35 36 36 be the conical points of S/H, then (S, H) and (C(λ ,...,λ − ), H ) are holomorphically equivalent and the 37 1 n 2 0 37 action of H is defined by the projective linear action H < PGL(n + 1, C).Wesaythat(S, H) is modeled by 38 0 38 the algebraic curve C(λ ,...,λ − ). 39 1 n 2 39 40 40 41 The structure of equations in (3) and Theorem 5 provides the following. 41 42 42 43 Corollary 6. A generalized Fermat curve of type (k,n) is the fiber product of (n − 1) classical Fermat curves of 43 44 type (k, 2). 44 45 45 46 Remark 7. Note that there are (n + 1)! choices in the normalization of the conical points as described 46 47 47 in Theorem 5. This normalization provides a canonical action of the symmetric group Sn+1 on the 48 48 locus of ordered tuples (λ1,...,λn−2). We will come back to this discussion in Section 4. 49 UNCORRECTED PROOF 49 50 50 Corollary 8. If (C(λ1,...,λn−2), H0) is a generalized hyperbolic Fermat pair of type (k,n),thenAutH (C(λ1, 51 0 51 ...,λn−2))/H0 is isomorphic to the subgroup of Möbius transformations that preserves the finite set 52 52 53 53 54 {∞, 0, 1,λ1,λ2,...,λn−2}. 54

1 1 Corollary 9. If (C(λ1,...,λn−2), H0) is a generalized hyperbolic Fermat pair of type (k,n),thenAutH0 (C(λ1 2 ,...,λn−2)) < PGL(n + 1, C). 2 3 3 4 Proof. Let (S, H) be a generalized Fermat curve. As consequence of Theorem 5 we may assume 4 5 5 (S, H) = (C(λ1,...,λn−2), H0).LetT be a Möbius transformation that permutes the conical points 6 6 μ1 =∞, μ2 = 0, μ3 = 1, μ4 = λ1, ..., μn+1 = λn−2. Corollary 8 asserts the existence of a conformal 7 7 automorphism T of C = C(λ1,...,λn−2) so that π T = T π ,thatis,ifT ([x1 :···: xn+1]) =[y1 :···: 8 k k 8 yn+1] and z =−(x2/x1) ,thenT (z) =−(y2/y1) . As the only cyclic subgroups of H or order k acting 9 9 with ﬁxed points are a j ,for j = 1,...,n + 1, there is a permutation σ ∈ Sn+1 (the symmetric group 10 −1 10 on (n + 1) letters) so that T a jT =aσ ( j). Now, since the zeros and poles of the meromorphic 11 11 function x j/x1 : C → C are Fix(a j ) and Fix(a1) respectively, we see that the zeros and poles of the 12 ∗ 12 pullback function T (x /x ) = (x /x ) ◦ T : C → C are Fix(a −1 ) and Fix(a −1 ) respectively. It fol- 13 j 1 j 1 σ ( j) σ (1) 13 ∈ −{ } ∗ = 14 lows that there exist c2,...,cn+1 C 0 such that T (x j /x1) c j(xσ −1( j)/xσ −1(1)) on C. This means 14 15 that in the open set {x1 = 0} the expression of the automorphism T in terms of aﬃne coordinates is 15 16 16 17 = 17 T (x2/x1,...,xn+1/x1) (c2xσ −1(2)/xσ −1(1),...,cn+1xσ −1(n+1)/xσ −1(1)). 18 18 19 Therefore, in projective coordinates T must be of the form 19 20 20 21 21 [ : :···: ] =[ − : − :···: − ] 22 T x1 x2 xn+1 xσ 1(1) c2xσ 1(2) cn+1xσ 1(n+1) . 22 23 23 = 24 The constants c j can be easily computed from the algebraic equations (3). Now, as T (z) 24 − k − − k =∞ = = = = − 25 c2(xσ 1(2)/xσ 1(1)) ,ifwesetμ1 , μ2 0, μ3 1andμ j+3 λ j ,for j 1,...,n 2, then 25 26 26

27 T (μ j ) = μσ ( j), 27 28 28 29 that is, the transformation T induces the same permutation of the index set {1,...,n + 1} as T . 29 30 This permits to compute easily AutH (S) (this can be implemented into a computer program) and 30 31 also to see that it is a subgroup of PGL(n + 1; C) whose matrix coordinates belong to the field 31 2π i/k 32 Q(e , c2,...,cn−2). 2 32 33 33 34 Remark 10. The procedure described in the above proof permits to define for each Möbius transfor- 34 35 35 mation T that permutes the conical points {∞, 0, 1,λ1,...,λn−2} an element T ∈ PGL(n + 1, C).Such 36 36 element is not uniquely defined by T , but it is unique up to composition with an element of H0.Also, 37 37 this representation satisfies that T1 T2 = T1 T2 (up to composition with some element of H0). 38 38 39 In the following examples we clarify the procedure given in the proof of Corollary 9. 39 40 40 41 2 41 Example 11. Let us clarify the above with the following example. Consider n = 4 and λ2 = λ = 42 1 42 1 + λ , then we obtain a generalized Fermat curve S of genus g = (2 − 5k3 + 3k4)/2. If k = 2, then 43 1 k 43 S corresponds to the classical Humbert curve of genus 5 with 160 conformal automorphisms. In this 44 44 case, the finite Möbius group that permutes the conical values ∞,0,1,λ and λ is the dihedral 45 1 2 45 group D generated by u(z) = λ /(λ − z) (permuting cyclically ∞,0,1,λ and λ )andv(z) = λ − z 46 5 2 2 1 2 2 46 (fixing ∞ and permuting 0 with λ2 and 1 with λ1). Liftings of u and v are given by 47 47 48 48 [ UNCORRECTED: : : : ] = : 1/k : : − PROOF1/k : − 1/k 49 u x1 x2 x3 x4 x5 x5 λ2 x1 x2 ( λ1) x3 ( λ2) x4 , 49 50 50 v [x : x : x : x : x ] = x : x : (−1)1/kx : (−1)1/kx : x . 51 1 2 3 4 5 1 5 4 3 2 51 52 52 5 = 2 = − 1/k − 1/k 1/k 1/k 53 Note that u v 1. Moreover, if we choose branches of ( λ2) , ( 1) , λ2 and λ1 so 53 1/k 1/k 1/k 54 − 1/k − 1/k = 2 = 2 = =∼ 54 that they all satisfy ( λ2) ( 1) λ2 and (λ1 ) λ2 ,then(u v) 1, that is, D5 u, v <

4 1 AutH (S). It follows that AutH (S) = H D5,inparticular,|AutH (S)|=10k . By Hurwitz’s upper bound, 1 2 if k = 2, then Aut(S) = AutH (S). 2 3 3 4 ={ k + k + k = }⊂P2 = 4 Example 12. If S x1 x2 x3 0 is a classical Fermat curve of degree k 4, that is, n 2, 5 ∼ 2 ∼ 2 5 then Aut(S) = Z S3 [27,30]. An easy way to see this is as follows. Let H = Z be a generalized Fer- 6 k k 6 mat group of type (k, 2) inside Aut(S). The quotient orbifold S/H has signature (0, 3;k,k,k). Clearly, 7 7 S/H admits the symmetric group S3 as group of orbifold automorphisms. A lift of S3 is generated by 8 8 the permutations of the coordinates: σ ([x1 : x2 : x3]) =[x2 : x1 : x3] and τ ([x1 : x2 : x3]) =[x3 : x1 : x2]. 9 9 In this case AutH (S) = H S3 and S/AutH (S) = (S/H)/S3 has signature (0, 3; 2, 3, 2k), which is 10 10 maximal if k 4 [28], in particular, Aut(S) = AutH (S). 11 11 12 12 13 3. On the uniqueness of generalized Fermat groups 13 14 14 15 3.1. Uniqueness for fixed type 15 16 16 17 In this section we are concerned with the uniqueness of the generalized Fermat group, within a 17 18 fixed type, on a fixed generalized Fermat curve. We are not able to provide a general answer to this 18 19 problem, but in the case of types (p,n), where p is a prime, we have the following partial result. 19 20 20

21 Theorem 13. Let p 2 be a prime and n 2 an integer so that (n − 1)(p − 1)>2.IfH1 and H2 are 21 22 generalized Fermat groups of type (p,n) for the same Riemann surface S, then they are conjugate in Aut(S). 22 23 23 24 Proof. Let H be a generalized Fermat group of type (p,n), where p 2 is a prime so that (n − 1)(p − 24 25 1)>2. We have a chain of p-subgroups 25 26 26 27 27 28 H = K0 ¡ K1 ¡ K2 ¡ ···¡ Kk 28 29 29 ∼ 30 where Kk is a p-Sylow subgroup and K j+1/K j = Zp .Ifk = 0, then we are done. Otherwise, Lemma 17 30 31 below asserts that K0 is unique in K1,henceK0 is also normal subgroup in K2, then (again by 31 32 Lemma 17) unique in K2. Proceeding in this way we end with K0 being unique in Kk. 2 32 33 33 34 Corollary 14. If n 2 is an integer and p 2 is a prime so that (n − 1)(p − 1)>2, then any Fuchsian 34 35 group of type (p,n) is uniquely determined by its commutator subgroup up to conjugation by some isometry 35 36 of hyperbolic plane. In particular, any two hyperbolic orbifolds of signature (0,n +1; p,...,p) are conformally 36 37 equivalent if and only if their homology cover Riemann surfaces are conformally equivalent. 37 38 38 39 39 In [15] (and for many classes of torsion free non-elementary Kleinian groups in [16–19,23]) was 40 40 noted that any Fuchsian group of type (∞,n), n 2, is uniquely determined by its commutator sub- 41 41 group. A natural question is if there is some integer q(n) so that if k q(n), then generalized Fermat 42 42 groups of type (k,n) should be unique in Aut(S). Related to this problem, the following was proved 43 43 in [21]. 44 44 45 45 46 Theorem 15. (See [21].) If n 2, then there exists a prime q(n) so that for each prime p q(n) the generalized 46 47 Fermat group of type (p,n) is unique in Aut(S). 47 48 48 49 Unfortunately, we doUNCORRECTED not have an explicit formula for q(n) and PROOFn large. It was observed in [9] that 49 50 a generalized Fermat group of type (2,n),forn = 4, 5, of a closed Riemann surface S is unique in 50 51 Aut(S), in particular, a normal subgroup of Aut(S). All the above permits to state the following fact. 51 52 52 53 Corollary 16. Let (S, H) be a generalized hyperbolic Fermat pair of type (p,n),wherepisaprime,andlet 53 54 q(n) as in Theorem 15.Ifeither 54

1 (i) p q(n),or 1 2 (ii) (p,n) = (2,n),wheren= 4, 5, 2 3 3 4 then Aut(S)/H is isomorphic to the subgroup of Möbius transformations preserving the finite set of conical 4 ∼ 5 points of S/H = C. 5 6 6 7 Lemma 17. Let p,n 2 be so that p is a prime and (n − 1)(p − 1)>2.LetL< Aut(S) be any p-subgroup of 7 8 conformal automorphisms of S. If L contains as normal subgroup a generalized Fermat group H of type (p,n), 8 9 then H is the unique generalized Fermat group of type (p,n) contained in L. 9 10 10 11 11 Proof. The case p = 2 was proved in [9]. From now on, we assume p 3 a prime. We use induction 12 12 on n. 13 13 If n = 2 and p 5, then S is (conformally equivalent to) the classical Fermat curves {xp + y p + zp = 14 2 14 0}⊂P , whose full group of conformal automorphisms is H S3 [27,30] (see also Example 12), in 15 15 particular, H = L. 16 16 Let n = 3, p 3 and H ¡ L as in the hypothesis. In this case, L/H is a finite p-group of orbifold 17 17 automorphisms of S/H. As (i) L/H is, in particular, a group of conformal automorphisms of C, (ii) the 18 18 finite subgroups of PSL(2, C) are cyclic groups, dihedral groups, alternating groups A4, A5 and the 19 19 symmetric group S4 [24] and (iii) p 3, it follows that H/L is either trivial (then H = L and we are 20 ∼ 20 done) or a cyclic group of order p. Let us assume L/H =τ = Zp .Asτ must permute the 4 conical 21 21 points, it follows that this case is only possible if p = 3. In this case, up to a Möbius transformation, 22 2π i/3 22 we may assume the conical points are ∞, 0, 1 and 1 + w3 (where w3 = e ), and τ (z) = w3z + 1. 23 23 It follows that S is given by 24 24 25 25 x3 + x3 + x3 = 0 26 1 2 3 , 26 + 3 + 3 + 3 = 27 (1 w3)x1 x2 x4 0. 27 28 28 29 In this case (see also Remark 20), L is generated by H and (assuming τ fixes the conical point of 29 30 S/H determined by the fixed points of a1([x1 : x2 : x3 : x4]) =[w3x1 : x2 : x3 : x4]) 30 31 31 32 1/3 32 α [x1 : x2 : x3 : x4] = (1 + w3) x1 : x4 : x2 :−x3 . 33 33 34 34 Note that α3 ∈ H −{I}. Now, direct computations permits to see that in L the generalized Fermat 35 35 group is unique. In this case, A4 =τ (z) = w3z + 1, η(z) = (z − 1)/z is the group of orbifold auto- 36 36 morphisms of S/H. As the 4 conical points of S/H arefixedpointsofelementsoforder3inA3 (the 37 37 4 vertices of a spherical tetrahedron in C invariant under A4) it follows that (S/H)/A4 has signature 38 38 (0, 3; 2, 3, 9) (the conical point of order 9 is the projection of one of the fixed point of τ , that is ∞, 39 39 the conical point of order 3 is the projection of the other fixed point of τ , that is 0, and the conical 40 40 point of order 2 is the projection of a fixed point of the involution τ 2η). As the signature (0, 3; 2, 3, 9) 41 41 is a maximal one [28], it follows that Aut(S) = AutH (S) =H, α,β, where 42 42 43 43 1/3 1/3 2 44 β [x1 : x2 : x3 : x4] = x2 : (1 + w3) x1 : x4 : (1 + w3) x3 β = I . 44 45 45 46 Let us assume our proposition to be true for types (p,m), where m n − 1 and n 4. 46 47 Let us assume also that we have a generalized Fermat group H < L,oftype(p,n) such that H = H. 47 48 Set G =H, H=H H and R = H ∩ H < Z(G) (where Z(G) denotes the center of G). Clearly, H ¡ G. 48 UNCORRECTED PROOF =∼ Zl 49 Let us consider the group G/H of conformal automorphisms of the orbifold S/H.AsG/H p ,for 49 50 some l 1, and S/H has underlying Riemann surface the Riemann sphere, necessarily l = 1, that is, 50 ∼ n+1 n−1 51 G/H = Zp . It follows that G has order p and, as |G|=|H||H|/|R|, that R has order p ,thatis, 51 =∼ Zn−1 2 52 R p . This also asserts that G/R is a group of order p ,thenanAbeliangroup.Wemaywrite 52 53 H =R, x and H =R, y, for some x ∈ H − R and y ∈ H − R (both of them then have order p). As 53 − − 54 G/R is Abelian, it follows that xyx 1 y 1 ∈ R and, in particular, that H ¡ G. 54

1 If R contains some standard generator, say t ∈ R, then we consider the generalized Fermat curve 1 2 S/t (the underlying Riemann surface structure) and the generalized Fermat groups H/t and H/t, 2 3 both of type (p,n − 1). By the inductive hypothesis, H/t=H/t;thenH = H, a contradiction. This 3 4 fact together with Parts 2-(iii) and 2-(iv) of Corollary 2 implies that R acts freely on S.SetM = S/R. 4 5 On M we have two commuting conformal automorphisms, say x and y,bothoforderp, which are 5 6 induced by x and y, respectively. The quotients of M by any of them is the Riemann sphere with 6 7 exactly (n + 1) conical points of order p; that is, each of these automorphisms has exactly (n + 1) 7 8 ﬁxed points. 8 9 By Proposition 1.8, in [12], the surface M is one of the following ones: 9 10 10 11 11 : p = p − 12 F p w u 1, 12 − 13 p p p p p 1 p 13 D p : w = u − 1 u − λ , some λ ∈ C, λ = 1, 14 14 15 15 16 and 16 17 17 18 x(u, w) = ηau, ηb w , 18 19 19 c d 20 y(u, w) = η u, η w 20 21 21 22 where 1 a, b, c, d p and η = e2π i/p . In there it is also noted that both automorphisms must have 22 23 23 either p (if M = F p )or2p (if M = D p ) ﬁxed points, that is, either n + 1 = p or n + 1 = 2p (it follows 24 that S/G has signature either (0, 3; p, p, p) or (0, 4; p, p, p, p)). 24 25 25 In the F p case, the conical points in S/H are exactly p and they are invariant under some elliptic 26 transformation u of order p (induced by y). We may assume, up to a Möbius transformation, that 26 27 27 these conical points are ∞,0,1,λ1,...,λp−3 and that u(∞) = 0, u(0) = 1, u(1) = λ1, u(λ j) = λ j+1 28 28 (for j = 1,...,p − 4) and u(λp−3) =∞. In this way, we can assume S = C(λ1,...,λp−3) and H = 29 ∼ p−1 29 H0 = Z is generated by standard generators a1,...,ap−1 (as described in Section 2; the other 30 p 30 standard generator is ap = a1 ···ap−1). A lifting of u, under the natural projection π[x1 :···: xp]= 31 p 31 −(x2/x1) , has the form u[x1 : ··· : xp]=[xp : c2x1 : ··· : cp xp−1], where c2,...,cp ∈ C −{0} (see 32 32 Corollary 9). All other liftings are of the form hu, where h ∈ H.Inthisway,y = hu for suitable h ∈ H. 33 33 As h ∈ H and y commute with each r ∈ R, this commutativity property also holds for each of the 34 34 = 2 3 ··· p−1 35 liftings u. Now, the only elements of H commuting with u are the powers of β a1a2a3 ap−1.In 35 [ :···: ] =[ : :···: ] 36 fact, a non-trivial element of H is of the form α( x1 xp ) w1x1 w2x2 w p xp , where 36 p = = 37 w j 1. We may assume w p 1. As 37 38 38 39 39 αu [x1 :···:xp] =[w1xp : w2c2x1 : w3c3x3 :···: w p−1cp−1xp−2 : cp xp−1], 40 40 41 41 uα [x1 :···:xp] =[xp : w1c2x1 : w2c3x3 :···: w p−2cp−1xp−2 : w p−1cp xp−1] 42 42

43 p−1 43 it follows that w = w2, w = w3, ..., w − = w , in particular, the above assertion. Since R has 44 2 1 3 1 p 1 1 44 n−1 45 order p there are plenty of elements in R which cannot commute with u, a contradiction. 45 [ :···: ]=[ : 46 In the D p case, the arguments are similar but in that case we should use u x1 x2p xp 46 :···: : : :···: ] ∈ C −{ } 2 47 c2x1 cp xp−1 cp+1x2p cp+2xp+1 c2p x2p−1 , where c j 0 . 47 48 48 49 Remark 18. If p is aUNCORRECTED prime and either (i) p > n + 1 or (ii) PROOFn + 1isnotcongruentto0,1or2 49 50 module p, then a generalized Fermat group of type (p,n) is a p-Sylow subgroup of Aut(S).Infact, 50 51 if H is a generalized Fermat group of type (p,n),thenH is either a p-Sylow subgroup of Aut(S) 51 52 or it is a normal subgroup of index p of some subgroup K < Aut(S). The last case will provide an 52 53 elliptic Möbius transformation of order p that preserves n + 1 points on the Riemann sphere S/H,a 53 54 contradiction to the conditions (i) or (ii). 54

1 Proposition 19. Let S be a generalized Fermat curve of type (p,n).Ifp 5 is a prime, then a generalized 1 2 Fermat group of type (p,n) cannot belong to two different p-Sylow subgroups of Aut(S). In particular, the 2 3 number of different generalized Fermat groups in S is equal to the number of p-Sylow subgroups in Aut(S). 3 4 4 5 Proof. Lemma 17 asserts that inside every p-Sylow subgroup of Aut(S) there is a unique generalized 5 6 Fermat group of type (p,n). Assume p 5isprime.IfH is a generalized Fermat group contained in 6 7 7 two different p-Sylow subgroups, say G1 and G2,thenS/H will admit two different automorphisms 8 of order p, both of them permuting the conical points; this would imply that there is a finite group of 8 9 Möbius transformations generated by two different transformations of order p 5, a contradiction (all 9 10 Z2 10 finite subgroups of Möbius transformations are either cyclic, 2, dihedral groups, alternating groups 11 11 A4 or A5 or the symmetric group S4 [24]). 2 12 12 13 13 Remark 20. Proposition 19 is false for p = 3. In fact, let us consider the generalized Fermat curve of 14 14 type (3, 3) 15 15 16 16 17 x3 + x3 + x3 = 0, 17 S = 1 2 3 18 + 3 + 3 + 3 = 18 (1 w3)x1 x2 x4 0. 19 19 20 = 20 21 We already noted in the proof of Lemma 17 that Aut(S) H0, α,β , where 21 22 22 23 1/3 9 3 23 α [x1 : x2 : x3 : x4] = (1 + w3) x1 : x4 : x2 :−x3 α = I, α ∈ H0 −{I} , 24 24 1/3 1/3 2 25 β [x1 : x2 : x3 : x4] = x2 : (1 + w3) x1 : x4 : (1 + w3) x3 β = I . 25 26 26 27 27 The generalized Fermat group H0 =a1,a2 is contained in the following two different 3-Sylow 28 28 subgroups of Aut(S): G1 =H0, α and G2 =H0,δ, where 29 29 30 30 [ : : : ] = : 1/3 :− : + 1/3 3 = 31 δ x1 x2 x3 x4 x4 w3 x1 x3 (1 w3) x2 δ 1 , 31 32 32 β = δα2 (up to an element of H ). 33 0 33 34 34 35 3.2. On the uniqueness of the type 35 36 36 37 In this section we provide a partial discussion on the problem about the uniqueness of the type on 37 38 a fixed Riemann surface S admitting a generalized Fermat group. Let us assume we have a closed Rie- 38 39 mann surface S which is a generalized Fermat curve of type (k,n). As already noted at the beginning, 39 40 the genus of S is of the form 40 41 41 42 φ(k,n) 42 g = 1 + (4) 43 2 43 44 44 − 45 where φ(k,n) = kn 1((n − 1)k − n − 1). The hyperbolicity on S ensures that the possible pairs (k,n) 45 46 belong to the set 46 47 47 48 48 49 UNCORRECTEDD = (k,n): k,n ∈{2, 3, 4,...},(k − 1)( PROOFn − 1)>2 . (5) 49 50 50 51 If we set n(2) = 4, n(3) = 3 and n(k) = 2, for k 4, then we easily see that 51 52 52 53 (P1) for each fixed k 2 and n n(k), we have that φ(k,n) is strictly increasing in n; 53 54 (P2) for each n 2 and k 2 so that n n(k), we have that φ(k,n) is strictly increasing in k. 54

1 The function φ is injective for most fixed values of g 2. The first two values of g for which 1 2 φ fails to be injective are g = 10 and g = 55, in which case, the possible values of (k,n) are (3, 3), 2 3 (6, 2) and (3, 4), (12, 2), respectively. However as the next proposition shows the uniqueness of type 3 4 is maintained in these two cases. 4 5 5 6 Proposition 21. No Riemann surface of genus 10 (respectively, 55) is simultaneously a generalized Fermat 6 7 curve of types (3, 3) and (6, 2) (respectively, (3, 4) and (12, 2)). 7 8 8 9 9 Proof. Assume we have a closed Riemann surface S,ofgenusg = 10, so that H H Aut S are 10 1, 2 < ( ) 10 generalizedFermatgroupsoftypes 3 3 and 6 2 , respectively. As we have that S H is an orbifold 11 ( , ) ( , ) / 2 11 of signature 0 3; 6 6 6 , it follows that S is the classical Fermat curve S ={x6 + x6 + x6 = 0}⊂P2.It 12 ( , , , ) 1 2 3 12 = Z2 13 is well known that Aut(S) 6 S3 [27,30] (see also Example 12) and that S/Aut(S) has signature 13 ; Z2 Z3 14 (0, 3 2, 3, 12). But, a group 6 S3 admits no subgroup isomorphic to 3, a contradiction. 14 15 Next, assume we have a closed Riemann surface S,ofgenusg = 55, so that H1, H2 < Aut(S) are 15 16 generalized Fermat groups of types (3, 4) and (12, 2), respectively. As we have that S/H2 is an orb- 16 ; ={ 12 + 12 + 12 = 17 ifold of signature (0, 3 12, 12, 12), it follows that S is the classical Fermat curve S x1 x2 x3 17 18 }⊂P2 = Z2 18 0 . It is well known that Aut(S) 12 S3 [27,30] (see also Example 12) and that S/Aut(S) has 19 ; Z2 Z4 19 signature (0, 3 2, 3, 24). But, a group 12 S3 admits no subgroup isomorphic to 3, a contradic- 20 tion. 2 20 21 21 22 22 Proposition 21 makes us wonder for the existence of a closed Riemann surface admitting general- 23 23 ized Fermat groups of different type. A partial answer to this question is provided by the following. 24 24 25 25 26 Proposition 22. If S is a hyperbolic generalized Fermat curve of even genus of types (k,n) and (k,n),sothat 26 = == 27 kandkareeven,thenk kandn n 2. 27 28 28 29 This is consequence of Lemma 23 below and property (P2) which says that φ(k, 2) is strictly 29 30 increasing in k. 30 31 31 32 Lemma 23. If S is a hyperbolic generalized Fermat curve of genus g and type (k,n), where k and g are even, 32 33 then n = 2 and k is not divisible by 4. 33 34 34 35 35 Proof. As in g = 2 we have no generalized Fermat curves, we must assume g 4. Let us write 36 36 37 37 38 − = a1 ··· ar 38 g 1 p1 pr , 39 39 40 40 where p j 3 are prime integers and a j 1. In this way, we have that a type (k,n) for this genus g 41 must satisfy from (4) the equality 41 42 42 43 43 − a 44 n 1 − − + = 1 ··· ar 44 k (n 1)k (n 1) 2p1 pr . 45 45 46 As we are assuming k even, the above obligates to have 46 47 47 48 48 − UNCORRECTEDn 1 = b1 ··· br PROOF 49 k 2p1 pr , 49 50 − − 50 − − + = a1 b1 ··· ar br 51 (n 1)k (n 1) p1 pr , 51 52 52 53 for certain values 0 b j a j .Asallprimesp j are different from 2, the first equality obligates to have 53 54 n = 2 and that k is not divisible by 4. 2 54

1 4. Moduli of generalized Fermat pairs 1 2 2 3 4.1. Some generalities 3 4 4 5 We now recall several known facts concerning parametrization of Riemann surfaces endowed with 5 6 a group action such as ours [13], see also [14]. Let S be closed Riemann surface of genus g and H < 6 0 0 7 Aut(S0). There is an irreducible complex analytic space M(H0) whose points are in bijection with the 7 8 isomorphy classes of pairs (S, H) topologically conjugate to (S0, H0). This space is the normalization 8 9 of the locus in the moduli space of genus g,sayMg , consisting of the classes of Riemann surfaces 9 10 admitting a group of conformal automorphisms topologically conjugate to H0.LetTg the Teichmüller 10 11 space of genus g and Modg the corresponding mapping class group of genus g. Then, Mg = Tg /Modg . 11 12 Let us consider H0 as a subgroup of Modg and let Tg (H0) ⊂ Tg be its locus of ﬁxed points. It is well 12 13 known that Tg (H0)/N(H0) = M(H0), where N(H0) denotes the normalizer of H0 inside Modg . 13 14 14 15 4.2. Generalized Fermat curves 15 16 16

17 Let us now assume (S0, H0) is a generalized Fermat pair of type (k,n) and genus g = gk,n, where 17 18 (k,n) is so that (n − 1)(k − 1)>2. As any two generalized Fermat pairs are topologically conjugate, 18 19 by Theorem 1, M(H ) contains the isomorphy classes of all generalized Fermat pairs of type (k,n). 19 0 20 We denote M(H0) simply by M(k,n) and by F(k,n) ⊂ Mg the locus consisting of those classes 20 21 [S]∈Mg where S is a generalized Fermat curve of type (k,n). The open connected set 21 22 22 23 n−2 n−2 23 Pn = (λ1,...,λn−2) ∈ C : λ j = 0, 1,λj = λi ⊂ C 24 24 25 is a model for the moduli space of ordered (n + 1) pointed sphere. Next we proceed to recall 25 26 the construction of a model of moduli space of unordered (n + 1) pointed sphere. We say that 26 27 27 (λ1,...,λn−2), (μ1,...,μn−1) ∈ Pn are equivalent if there is a Möbius transformation A ∈ PSL(2, C) 28 so that A sends the set 28 29 29 30 30 {∞ } 31 , 0, 1,λ1,λ2,...,λn−2 31 32 32 33 onto 33 34 34

35 {∞, 0, 1, μ1, μ2,...,μn−2}. 35 36 36

37 The above equivalence relation is in fact given by action of the symmetric group Sn+1 on n + 1 37 38 letters as follows. For each σ ∈ Sn+1 and each (λ1,...,λn−2) ∈ Pn we form the (n +1)-tuple (x1 =∞, 38 39 x2 = 0, x3 = 1, x4 = λ1,...,xn+1 = λn−2). Then, we consider the (n + 1)-tuple (xσ (1),...,xσ (n+1)) 39 40 and let Tσ ∈ PSL(2, C) be the unique Möbius transformation so that Tσ (xσ (1)) =∞, Tσ (xσ (2)) = 0 40 41 and Tσ (xσ (3)) = 1. We set μ j = Tσ (xσ ( j+3)),for j = 1,...,n − 2. In this way, we obtain that 41 42 (μ1,...,μn−1) ∈ Pn and an action of Sn+1 on Pn. The quotient space Qn obtained by this action 42 43 is a model of the moduli space M0,n+1 of the unordered (n + 1) pointed sphere. 43 44 The space F(k,n) is related to the moduli space Qn introduced above, as follows. Let us denote by 44 45 [S] (resp. [S, H], [S/H]) the isomorphy class of the Riemann surface S (resp. the pair (S, H), and the 45 46 orbifold S/H), then the rule that sends [S, H] to [S] (resp. to [S/H]) deﬁnes a surjective holomorphic 46 47 mapping π1 : M(k,n) → F(k,n) ⊂ Mg (resp. π2 : M(k,n) → Qn). Moreover, π1 is known to be a 47 48 ﬁnite surjective mappingUNCORRECTED which fails to be injective if and only PROOF if there are surfaces S admitting two 48 49 generalized Fermat groups H1 and H2,bothoftype(k,n), which are not conjugate within the full 49 50 group Aut(S); so that, [S, H1] and [S, H2] are different points of M(k,n) which map to the same 50 51 point [S]∈F(k,n) ⊂ Mg .Asforπ2, Theorem 1 tells us that it induces an isomorphism between Qn 51 52 and M(k,n). We can now state the following results. 52 53 53

54 Theorem 24. The moduli space Qn deﬁnes a generically one-to-one cover of F(k,n). 54

1 Proof. As explained above, injectivity of π1 : Qn → Mg only fails at points corresponding to Fermat 1 2 curves S such that Aut(S) contains a subgroup G generated by two non-conjugate Fermat groups 2 3 H1, H2 of same type (k,n).SinceG strictly contains H1, we see that S/G is an orbifold of genus zero 3 4 with r < n + 1 branched points. This implies that the locus of unwanted points is a finite union of 4 5 spaces of dimension r − 3withr < n + 1, hence its dimension is strictly lower than that of Qn. 2 5 6 6 7 7 The space Qn is the normalization of F(k,n). In the particular case that k = p is prime, the 8 argument given to prove Theorem 24 combined with Theorem 13 clearly implies the following more 8 9 precise statement. 9 10 10 11 11 Theorem 25. Let p,n > 0 be integers with (n − 1)(p − 1)>2 and p is a prime. Then, Qn is isomorphic to 12 F(p,n), that is, F(p,n) is a normal variety. 12 13 13 14 14 Corollary 26. Let p,n > 0 be integers with (n − 1)(p − 1)>2 and p a prime. Then, the moduli space of 15 ∼ 15 n + 1 unordered points in the sphere (= Qn) embeds holomorphically inside the moduli space Mg ,where 16 16 g = 1 + φ(p,n)/2, as the locus of generalized Fermat curves of type (p,n). 17 17 18 18 5. Applications of generalized Fermat curves 19 19 20 20 In this section we provide three higher-dimensional applications of generalized Fermat curves. The 21 21 first one is the construction of a pencil of non-hyperelliptic Riemann surfaces with exactly three sin- 22 22 gular fibers. The second one is related to the previous one and consists of the definition of an injective 23 23 holomorphic map from C −{0, 1} into some moduli space M . The third one is the description of 24 g 24 all complex surfaces isogenous to a product X = S × S /G with invariants p = q = 0 and group G 25 1 2 g 25 equals either G = Z2 or G = Z4. 26 5 2 26 : → P1 27 The link between the first two applications is made as follows. If π X is a pencil of curves 27 ∈{ ∞} 28 of genus g with singular fibers at t 0, 1, , then, by the universal property of moduli space, 28 : C −{ }→M ∈ C −{ } 29 there is a holomorphic map ψ 0, 1 g defined by sending t 0, 1 to the point in 29 M −1 M 30 g representing the isomorphism class of the curve π (t).Conversely, g comes equipped with 30 : C → M ∈ M 31 a fiber space π g g , the universal curve, whose fiber over a point x g is generically the 31 32 Riemann surface Cx that this point represents, and so our initial pencil is obtained as pull-back of 32 C 33 g by ψ. However, this correspondence is not a perfect one because if x represents a curve with 33 : C −{ }→M 34 automorphisms the true fiber is Cx/Aut(Cx). Therefore, if a map ψ 0, 1 g ,suchastheone 34 35 we construct below, parametrizes Riemann surfaces with automorphisms it is not, in principle, clear 35 36 that it must be induced by a pencil of curves. 36 37 37 38 5.1. A pencil of generalized Fermat curves with three singular fibers 38 39 39 P1 40 It was observed by A. Beauville [4] that every family of curves over , with variable moduli, 40 41 admits at least 3 singular fibers. In that paper an example for which the minimal number of singular 41 42 fibers occurs is provided by certain family of hyperelliptic Riemann surfaces. Another example of 42 43 this kind was constructed by González-Aguilera and Rodríguez [11]; in this case all curves in the 43 A 44 family possess an automorphism group which contains the alternating group in 5 letters 5.Next 44 45 we provide another example of this sort in which the members of the family are generalized Fermat 45 46 curves, hence never hyperelliptic. 46 = P = C −{ } P1 × P3 47 Let us consider n 3 and k 3. In this case 3 0, 1 .Inside let us consider the 47 48 complex surface 48 49 UNCORRECTED PROOF 49 k + k + k = 50 x1 x2 x3 0, 1 3 50 X = : [a : b]∈P , [x1 :···:x4]∈P . 51 k + k + k = 51 ax1 bx2 bx4 0 52 52 53 Equipped with the projection into the first factor, this is a pencil whose fibers we denote by 53 54 V ([a : b]). 54

1 By taking b = 1 and a ∈ C −{0, 1}, we see that this is a moduli family of non-hyperelliptic curves 1 2 with variable moduli. We already known that the only singular fibers are provided by V ([0 : 1]), 2 3 V [1 : 1] and V ([1 : 0]). 3 4 The singular fiber V ([1 : 0]) ={[0 : 1 : w : z]∈P3 : wk =−1}∪{[0 : 0 : 0 : 1]} is given by k different 4 1 3 5 P ⊂ P all of them sharing a unique point [0 : 0 : 0 : 1]. The singular fiber V ([0 : 1]) ={[x1 : x2 : 5 6 : ]∈P3 k + k + k = = k =− } 6 x3 x4 : x1 x2 x3 0, x4 wx2, w 1 is given by k different classical Fermat curves of 7 order k all of them sharing exactly k points, these being [1 : 0 : w : 0], where wk =−1. Similarly, the 7 8 singular fiber V ([1 : 1]) is given by k different classical Fermat curves of order k all of them sharing 8 9 exactly k points, these being [1 : w : 0 : 0], where wk =−1. In particular, V ([0 : 1]) ≡ V ([1 : 1]). 9 10 10 11 5.2. An injective embedding of the thrice punctured sphere in moduli space 11 12 12 13 Let q(59) as in Theorem 15, enjoying the property that for every prime p q(59) we have that 13 14 a closed Riemann surface S has at most one generalized Fermat group of type (p, 59), which then 14 15 must be a normal subgroup of Aut(S). Let us choose once for all such a prime integer p q(59) and, 15 16 16 throughout this section, let us denote by gp the genus of the generalized Fermat curve of type (p, 59). 17 17 Now, let us regard A5 as a finite group of Möbius transformations and let π : C → C be a holo- 18 18 morphic branched covering map induced by the action of A5. Let us assume the branch values of π 19 −1 19 are given by ∞, 0 and 1. For each t ∈ C −{0, 1}, we consider the preimage π (t) ={μ1,...,μ60}. 20 We may then consider a Möbius transformation to send 3 of these preimages onto ∞, 0 and 1; 20 21 21 the others are denoted by λ1,...,λ57 . This gives us a generalized Fermat curve of type (p, 59),say 22 22 C(λ1,...,λ57) = C(t). The previous choices are not unique, but the conformal class [C(t)] of the re- 23 sulting generalized Fermat curve is uniquely determined by the value of t.Thisassertsthatthereisa 23 24 well defined holomorphic map 24 25 25 26 26 : C −{0 1}→F p 59 ⊂ M 27 ψ , ( , ) gp . 27 28 28 = 29 Assume ψ(t1) ψ(t2). In this case, the uniqueness of the generalized Fermat group, asserts the 29 −1 = −1 −1 ={ } 30 existence of a Möbius transformation T so that T (π (t1)) π (t2).Ifπ (t1) μ1,...,μ60 30 −1 ={ } A 31 and π (t2) μ1,...,μ60 , then as both sets are invariant by (the same) 5, it follows that T 31 A = A 32 normalizes 5.SetG T , 5 .AsT leaves invariant the set of fixed points of the order 3 elements 32 A 33 in 5 (20 points) and each Möbius transformation is uniquely determined by its action at three 33 34 different points, it follows that T has finite order and that G is finite. But, it is well known that 34 A 35 there is no finite subgroup of Möbius transformations containing 5 as a proper subgroup [24], in 35 ∈ A 36 particular, T 5. It follows that ψ is an injective map. 36 { } A 37 As the set λ1,...,λ60 is invariant under the action of 5, C(λ1,...,λ57) has the property that 37 = =∼ Z59 A∗ A∗ 38 Aut(C(λ1,...,λ57)) contains the group L H p , 5 , where 5 denotes the set formed by a 38 A 39 lifting of each element of the group 5, which exists by Corollary 2. As there is no finite group of 39 A = 40 Möbius transformations containing strictly 5, it follows that in fact L Aut(C(λ1,...,λ57)).Asa 40 ∈ C −{ } 41 consequence, for every t 0, 1 , the automorphism group is the same at each surface C(t). 41 42 All the above is summarized in the following. 42 43 43 44 Proposition 27. Let q(59) be as in Theorem 15 and p q(59) a prime number. Then the above construction 44 : C −{ }→F ⊂ M ∈ C −{ } 45 provides an injective holomorphic map ψ 0, 1 (p, 59) gp which sends t 0, 1 to the 45 M ∈ C −{ } 46 point in gp representing the generalized Fermat curve C(t). Moreover, for each t 0, 1 Aut(C(t)) is 46 A Z59 47 an extension of 5 by p . 47 48 48 49 Remark 28. Let us goUNCORRECTED back to the group L which arises as PROOF the full automorphism group of the 49 50 curves occurring in Proposition 27. In the language of Section 4.1 consider the quotient space 50 51 M = T 51 (L) gp (L)/N(L). By general results of Teichmüller theory [25] one knows that the stabilizer 52 ∈ T = ∩ 52 of a point x gp (L) is precisely the group Stab(x) Aut(x) N(L) where Aut(x) is the full group 53 of automorphisms of a suitable model Sx of the Riemann surface parametrized by the point x.This 53 54 T 54 fact has two implications. Firstly, we see that L acts trivially on gp (L), so we may as well write

1 M = T = ∈ T 1 (L) gp (L)/(N(L)/L). Secondly, since Aut(x) L for each x gp (L), we infer that N(L)/L acts 2 T 2 fixed point freely on gp (L).Moreover,sinceSx/L is an orbifold of genus zero with four branching 3 values (in fact, with signature (0, 4; 2, 3, 5, p)), it follows, again by general results of Teichmüller the- 3 4 T 4 ory, that gp (L) is isomorphic to the Teichmüller space of a 4-puncture sphere, which we may identify 5 with the hyperbolic plane H2. The conclusion is that M(L) is a Riemann surface and that N(L)/L is 5 6 its fundamental group. Now by Proposition 27 this Riemann surface contains C−{0, 1}.Hence,wecan 6 ∼ 7 only have M(L) = C −{0, 1} and so N(L)/L is isomorphic to Γ(2) = Z ∗ Z, the principal congruence 7 8 subgroup of level 2 of PSL(2, Z). 8 9 9 10 5.3. Complex surfaces isogenous to a product 10 11 11 12 By a complex surface we shall mean a compact holomorphic manifold of complex dimension two, 12 13 hence a real four manifold. 13 14 14 A complex surface X is said to be isogenous to a higher product if there are Riemann surfaces S1 15 15 and S2 of genus 2 and a finite group G acting freely on S1 × S2 by biholomorphic transformations 16 16 such that X is isomorphic to the quotient S1 × S2/G. If the action of G preserves each of the fac- 17 tors then X is said to be of unmixed type. In what follows we shall assume that this is always the 17 18 case. Surfaces isogenous to a higher product have been extensively studied by Bauer, Catanese and 18 19 Grunewald [3,7,8]. 19 20 The first example of a complex surface isogenous to a higher product was given by Beauville as 20 21 21 exercise number 4 in page 159 of [5]. In his example the algebraic curves S1, S2 are both the Fermat 22 5 + 5 + 5 = Z2 × 22 curve S: X1 X2 X3 0 and G is the group 5 acting on S S by, say, 23 23 24 24 [ : : ] [ : : ] = a : b : a+3b : 2a+4b : 25 (a, b) x1 x2 x3 , y1 y2 y3 ξ x1 ξ x2 x3 , ξ y1 ξ y2 y3 25 26 26 27 where ξ is 5th root of unity. 27 28 The results in this paper allow us to extend Beauville’s construction to generalized Fermat pairs 28 29 as follows. Let us consider two generalized Fermat pairs (S1, H1) and (S2, H2),bothoftype(k,n), 29 − − 30 where k,n 2, (k 1)(n 1)>2. 30 = ··· = 31 Let a1,...,an and an+1 a1 an be standard generators of H1 and let b1,...,bn and bn+1 31 ··· 32 b1 bn be standard generators of H2. 32 33 Remember that the only non-trivial elements of the generalized Fermat group H2 acting with fixed 33 + 34 points are contained in the cyclic groups generated by the n 1 standard generators b1,...,bn+1. 34 35 35 36 Remark 29. If n = 2 and k 4, a necessary and sufficient condition for the existence of generators 36 37 of H2 acting freely on S2 is that LCM(k, 6) = 1, where LCM stands for “least common multiple” (see 37 38 [7,8]). 38 39 39 40 If n = 2 (in which case k 4) and LCM(k, 6) = 1, a set of generators acting freely is given by 40 41 41 42 = 2 = −1 42 c1 b1b2, c2 b1b2 . 43 43 44 44 If n = 3, k 4 and LCM(6,k) = 1, then a set of generators of H2, each one acting freely on S2,is 45 given by 45 46 46 47 47 = 2 = −1 = 48 c1 b1b2, c2 b1b2 , c3 b1b3. 48 49 UNCORRECTED PROOF 49 = = 50 If n 3 k, then a set of generators of H2, each one acting freely on S2,isgivenby 50 51 51 52 = 2 = = 52 c1 b1b2, c2 b1b2, c3 b2b3. 53 53

54 If n 4, then a set of generators of H2, each one acting freely on S2,isgivenby 54

1 c1 = b1b2b3, c2 = b1b2, c3 = b2b3, c j = b1b j , j = 4,...,n. 1 2 2 3 3 Note that, in any of the above case, cn+1 = c1c2 ···cn also acts freely on S2. 4 : Zn → × 4 Let us consider an action Φ k Aut(S1 S2) as follows. If e1,...,en denotes the standard basis 5 Zn 5 of k ,thenset 6 6 7 7 Φ(e j) : S1 × S2 → S1 × S2, 8 8 9 9 Φ [x1 :···:xn+1], [y1 :···: yn+1] → a j [x1 :···:xn+1] , c j [y1 :···: yn+1] 10 10 11 Zn 11 and extend it to the rest of k in the natural way. 12 12 13 13 Proposition 30. Let (S1, H1) and (S2, H2) be two generalized Fermat pairs of same type (k,n),where(n − 14 − ∈{ } = : Zn → × 14 1)(k 1)>2.Ifn 2, 3 and k 4 we assume LCM(6,k) 1.LetΦ k Aut(S1 S2) be the action 15 n n 15 constructed above. Then, Φ(Z ) acts freely on S1 × S2 giving rise to a complex surface X = S1 × S2/Z with 16 k k 16 geometric genus p g and irregularity q = 0 where 17 17 18 18 n−2 − − − 2 19 k ((n 1)(k 1) 2) 19 p g = − 1. 20 4 20 21 21 22 Proof. That the action is free is consequence of the previous observations and construction. The fact 22 23 that the irregularity q vanishes is a consequence of the fact that each of the quotient Riemann surfaces 23 P1 24 Si /Hi is isomorphic to which does not admit non-zero holomorphic differentials (cf. [2], p. 303). 24 25 The statement relative to the geometric genus p g follows from the following identity (see [2,4,8]) 25 26 26 2 27 1 (gk,n − 1) 27 1 + p g − q = χ(X) = χtop(X) = 28 |Zn| 28 4 4 k 29 29 30 30 where χtop(S) stands for the Euler–Poincaré characteristic of X, χ(X) =: χ(OX ) for the Euler charac- 31 31 teristic of the structure sheaf of X and gk,n denotes the genus of a Fermat curve of type (k,n). 2 32 32 33 = × Zn = = 33 Corollary 31. The above construction gives complex surfaces X S1 S2/ k with invariants p g q 0 if 34 and only if (k,n) = (5, 2) or (k,n) = (2, 4). 34 35 35 Conversely, if X = S1 × S2/G is a complex surface isogenous to a higher product with invariants p g = q = 0 36 = Z2 = Z4 36 and group G 5 (resp. G 2)thenS1 and S2 are generalized Fermat curves of type (5, 2) (resp. (2, 4)). 37 37 38 38 Proof. As we are looking for complex surfaces with q = 0, both quotients S1/G and S2/G are isomor- 39 1 39 phic to P . Exploiting the Riemann–Hurwitz formula, it is shown in [3] that the orbifolds Si /G have 40 40 to have both signature (0, 3; 5, 5, 5) (resp. (0, 5; 2, 2, 2, 2, 2)), hence S1 and S2 are generalized Fermat 41 41 curves of the required type. 2 42 42 43 43 Remark 32. Bauer and Catanese [3] have proved that when G = Z2 one obtains exactly two non- 44 5 44 = Z4 45 isomorphic complex surfaces whereas for G 2 the complex surfaces so obtained form an irre- 45 46 ducible connected component of dimension 4 in their moduli space. The dimensions of these compo- 46 47 nents are in accordance with our Theorem 25 which implies that these components are parametrized 47 Q × Q Q × Q 48 by the space 3 3 (resp. 5 5)viatherule 48 49 UNCORRECTED PROOF 49

50 (S1, S2) → X = S1 × S2/G. 50 51 51 52 Furthermore, by the uniqueness of this representation of X (Proposition 3.13 in [8]), this map is 52 53 injective modulo the action of Z2 which interchanges the factors of Qn × Qn.Intheﬁrstcasethetwo 53 54 Z2 54 different components (of dimension 0) arise from two different actions of the group 5 whereas in

1 Z4 1 the second case the above seems to indicate that all possible actions of the group 2 are equivalent 2 and that the irreducible component in question is Q5 × Q5/Z2. 2 3 In [3] it is also shown that there are only two other Abelian groups G whichcangiveriseto 3 4 = × = = = Z3 = Z2 4 complex surfaces X S1 S2/G with invariants p g q 0, namely G 2 and G 3. 5 5 6 6. Connection with handlebodies 6 7 7 8 A Schottky uniformization of a closed Riemann surface S is a triple (Ω, G, p : Ω → S), where G is 8 9 a Schottky group with region of discontinuity Ω and p : Ω → S is a regular holomorphic covering 9 10 3 10 with G as covering group. The 3-manifold with boundary MG = (H ∪ Ω)/G is a handlebody body 11 of genus g whose interior admits a complete hyperbolic metric with injectivity radius bounded from 11 12 12 below by zero. Let t be a hyperbolic isometry of the interior of MG . By lifting it to the universal 13 cover H3, we obtain a Möbius transformation in the normalizer of the Schottky group G;soitdefines 13 14 naturally a continuous extension of t as a conformal automorphism of S. Reciprocally, a conformal 14 15 15 automorphism of S extends continuously as an isometry of the interior of MG if and only if it lifts to 16 Ω (by the given Schottky uniformization) as a conformal automorphism (this because the region of 16 17 17 discontinuity of a Schottky group is of class O AD). As a consequence, a group H < Aut(S) extends as 18 18 a group of isometries of the interior of MG if and only if H lifts under the Schottky uniformization as 19 a group of automorphisms of Ω. 19 20 20 21 Theorem 33. Let (S, H) be a generalized Fermat pair of type (k,n).ThenHcanbeextendedasgroupof 21 22 hyperbolic isometries of some handlebody whose conformal boundary is S if and only if k = 2. 22 23 23 24 24 Proof. Necessary and sufficient conditions for H Abelian to lift to some Schottky uniformization of 25 25 S are given in [20]. A necessary condition is that every non-trivial element of H must have an even 26 26 number of fixed points. Now, let H be a generalized Fermat group of type (k,n). The above together 27 27 with part (2.-(ii)) of Corollary 2 asserts that k should be even. Assuming k even, choose any element 28 28 a of the standard set of generators of H, and choose any two fixed points of it, say x and y.Part 29 29 (2.-(v)) of Corollary 2 asserts the existence of some h ∈ H so that h(x) = y, so the angles of rotation 30 30 of a1 at both x and y are the same. This violates the necessary conditions of [20] for H to lift to some 31 31 Schottky uniformization if k > 2. The case k = 2 it is known to be of Schottky type. 2 32 32 33 33 Uncited references 34 34 35 35 [1] 36 36 37 37 Acknowledgment 38 38 39 39 We would like to thank the referee for her/his valuable comments which improved the presenta- 40 40 tion of the paper and saved us from several mistakes. 41 41 42 42 References 43 43 44 44 [1] S.S. Abhyankar, Algebraic Space Curves, Les Presses de L’Université de Montreal, Montreal, Canada, 1971. 45 [2] Wolf P. Barth, Klaus Hulek, Chris A.M. Peters, Antonius Van de Ven, Compact Complex Surfaces, second edition, Ergeb. 45 46 Math. Grenzgeb. (3), vol. 4, Springer-Verlag, Berlin, 2004. 46 = = 47 [3] I. Bauer, F. Catanese, Some new surfaces with pg q 0, in: The Fano Conference, Univ. Torino, Turin, 2004, pp. 123–142. 47 [4] A. Beauville, Le nombre minimum de fibres singuliéres d’une courbe stable sur P1, in: Lucien Szpiro (Ed.), Séminaire sur 48 48 les pinceaux de courbesUNCORRECTED de genre au moins deux, Astérisque 86 (1981) 97–108. PROOF 49 [5] A. Beauville, Surfaces algébriques complexes, Astérisque 54 (1978). 49 50 [6] E. Bujalance, J.J. Etayo, E. Martinez, Automorphisms groups of hyperelliptic Riemann surfaces,KodaiMath.J.10(1987) 50 51 174–181. 51 52 [7] I. Bauer, F. Catanese, F. Grunewald, Beauville surfaces without real structures, I, in: F. Bogomolov, Y. Tschinkel (Eds.), Geo- 52 metric Methods in Algebra and Number Theory, in: Progr. Math., vol. 235, Birkhäuser, 2005, pp. 1–42. 53 53 [8] F. Catanese, Fibred surfaces, varieties isogenous to a product and related moduli spaces, Amer. J. Math. 122 (2000) 1–44. 54 [9] A. Carocca, V. González, R.A. Hidalgo, R. Rodríguez, Generalized Humbert curves, Israel J. Math. 64 (1) (2008) 165–192. 54

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Please cite this article in press as: G. González-Diez et al., Generalized Fermat curves, J. Algebra (2009), doi:10.1016/j.jalgebra.2009.01.002