A family of (p,n) -gonal Riemann surfaces with several (p,n) -gonal groups
aa r X i v : . [ m a t h . AG ] J un A FAMILY OF ( p, n ) -GONAL RIEMANN SURFACES WITHSEVERAL ( p, n ) -GONAL GROUPS SEBASTI ´AN REYES-CAROCCA
Abstract.
Let p > n > p − n. In this short note we construct a family of ( p, n )-gonalRiemann surfaces of maximal genus 2 np + ( p − with more than one ( p, n )-gonal group. Introduction and statement of the result
Let S be a compact Riemann surface of genus g > S ) denoteits automorphism group. If p > n > S is called ( p, n ) -gonal if there exists a group of automorphisms C p ∼ = H Aut( S )such that the corresponding orbit space S/H has genus n. The group H is calleda ( p, n ) -gonal group of S. Each compact Riemann surface with non-trivial automorphisms is ( p, n )-gonalfor suitable of p and n. This simple fact shows that to study ( p, n )-gonal Riemannsurfaces and their automorphisms is equivalent to study the singular locus ofthe moduli space of compact Riemann surfaces.( p, n )-gonal Riemann surfaces and their automorphisms have been extensivelyconsidered over the last century as they generalize important and well-studiedclasses of Riemann surfaces, such as (2 , hyperelliptic , ( p, p -gonal and (2 , n )-gonal or n -hyperelliptic Riemann surfaces, among others.Let S be a p -gonal Riemann surface of genus g > . By the classical Castelnuovo-Severi inequality (see Accola’s book [1]), if g > ( p − (1.1)then the p -gonal group is unique in the automorphism group of S . A familyof p -gonal Riemann surfaces of maximal genus g = ( p − endowed with two p -gonal groups was constructed in [2], showing that the bound (1.1) is sharp.Furthermore, in the general case, following [3], if S has two p -gonal groups then Mathematics Subject Classification.
Key words and phrases.
Compact Riemann surfaces, group actions, automorphisms.Partially supported by Fondecyt Grants 11180024, 1190991 and Redes Grant 2017-170071. they are conjugate in the automorphism group of S . An upper bound for thenumber of such groups was obtained in [4].For ( p, n )-gonality with n > , the Castelnuovo-Severi inequality ensures thatif S is a ( p, n )-gonal Riemann surface of genus g > g > pn + ( p − (1.2)then the ( p, n )-gonal group is unique in the automorphism group of S . In thegeneral case, it was proved in [5] that if S is a ( p, n )-gonal Riemann surfaceof genus g and p > n + 1 then all its ( p, n )-gonal groups are conjugate in theautomorphism group of S ; an upper bound for the size of the corresponding con-jugacy class was also determined in the same paper. Later, in [6], the uniquenessof the ( p, n )-gonal group was proved to be true under the assumptions that the( p, n )-gonal group acts with fixed points and p > n − . This short note is devoted to provide a family of ( p, n )-gonal Riemann surfacesof maximal genus g = 2 pn + ( p − with two ( p, n )-gonal groups. The existenceof this family shows that the bound (1.2) is sharp, for infinitely many pairs ( p, n ) . Theorem.
Let p > n > p − n. Set d = n/ ( p −
1) + 1 . Then there exists a complex d -dimensional family of ( p, n )-gonal Riemann sur-faces S of genus g = 2 np + ( p − with automorphism group of order 4 p acting on S with signature(0; 2 , , , p, d . . ., p )in such a way that each S has more than one ( p, n )-gonal group. Remark.
It is worth mentioning here the following observations which will followfrom the proof of the theorem.(1) The result remains true if p = 2 and n is odd.(2) If n = 0 our family agrees with the family constructed in [2].2. Proof of the Theorem
Let ∆ be a Fuchsian group of signature (0; 2 , , , p, d . . ., p ) canonically pre-sented as∆ = h γ , . . . , γ d +3 : γ = γ = γ = γ p = · · · = γ pd +3 = γ · · · γ d +3 = 1 i and consider the group G = D p × D p (where D p denotes the dihedral group oforder 2 p ) presented in terms of generators s , s , r , r and relations s = s = r p = r p = ( s r ) = ( s r ) = [ s , r ] = [ s , s ] = [ r , r ] = [ r , s ] = 1 . FAMILY OF ( p, n )-GONAL RIEMANN SURFACES 3
Existence of the family.
By virtue of the classical Riemann’s existence theo-rem, the existence of the desired family follows after verifying that the Riemann-Hurwitz formula holds and after providing a surface-kernel epimorphism θ from∆ onto G. Note that the equality2( g −
1) = 4 p ( − − ) + d (1 − p ))shows that the Riemann-Hurwitz formula is satisfied for a branched 4 p -foldcovering map from a compact Riemann surface of genus g = 2 np + ( p − onto the projective line, ramified over three values marked with 2 and d valuesmarked with p .In addition, if d is odd we can choose the surface-kernel epimorphism θ as θ ( γ ) = s , θ ( γ ) = s , θ ( γ ) = s s r r and θ ( γ i ) = (cid:26) ( r r ) − if i is even r r if i is odd,and if d is even we can choose θ as θ ( γ ) = s , θ ( γ ) = s , θ ( γ ) = s s ( r r ) − d/ and θ ( γ i ) = (cid:26) r if i is even r if i is oddwhere i ∈ { , . . . , d + 3 } . The complex dimension of the family agrees with the complex dimension ofthe Teichm¨uller space associated to ∆; namely, its dimension is d. ( p, n ) -gonal groups. We denote the branched regular covering map given bythe action of G on S by π : S → S/G and its branch values by y , y , y , z . . . , z d ,where each y k is marked with 2 and each z k is marked with p. Assume d odd. Consider the cyclic subgroups of order pH = h r r i and H = h r − r i of G. We denote by π and π the branched regular covering maps given bythe action of H and H on S respectively. We observe that the fiber of π overeach y k does not contain any branch value of π and π . In addition, for each k , the fiber of π over z k has 4 p elements; the isotropy group of 2 p of them isisomorphic to H and the remaining ones have isotropy group isomorphic H . It follows that π and π ramify over 2 pd values, each of them marked with p. Equivalently, the signature of the action of H j on S is ( n j ; p, dp . . ., p ) where n j isthe genus of S/H j . We now consider the Riemann-Hurwitz formula to see that2( g −
1) = p [2 n j − pd (1 − p )]and, after straightforward computations, one obtains that n j = n for j = 1 , d even. Consider the cyclic subgroups of order pH = h r i and H = h r i SEBASTI ´AN REYES-CAROCCA of G and let π and π be as before. As in the previous case, the fiber of π overeach y k does not contain any branch value of π and π . For each k the fiberof π over y k has 4 p elements; the isotropy group of them is isomorphic to H if k is odd and is isomorphic to H if k is even. It follows that π and π ramifyover 2 pd values, each of them marked with p. Equivalently, the signature of theaction of H j on S is ( n j ; p, dp . . ., p ) where n j is the genus of S/H j . Similarly aspreviously done, the Riemann-Hurwitz formula ensures that n j = n for j = 1 , H and H are two ( p, n )-gonal groups of S, as desired. Remark.
Note that if d is odd then the ( p, n )-gonal groups are conjugate, butif d even then they are not. References [1]
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On the uniqueness of ( p, h ) -gonal automorphisms of Riemann surfaces, Arch. Math. (Basel) (2012), no. 6, 591–598. Departamento de Matem´atica y Estad´ıstica, Universidad de La Frontera,Avenida Francisco Salazar 01145, Temuco, Chile.
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