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Generalized Fermat Curves 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 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.2 (1-18) 2 G. González-Diez et al. / Journal of Algebra ••• (••••) •••–••• 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<PSL(2, R), that is, H2/Γ = O. The group Γ has presentation (we say that Γ 53 54 is of type (k,n)) 54 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.
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