On symmetries of pq-hyperelliptic Riemann surfaces
DOI:
https://doi.org/10.4067/S0716-09172006000200004Keywords:
Automorphisms of Riemann surface, p-hyperelliptic Riemann surface, fixed points of automorphism, symmetry, automorfismos de superficie de Riemann, superficie de Riemann p-hiperelíptica, puntos fijos de automorfismo, simetría.Abstract
A symmetry of a Riemann surface X is an antiholomorphic involution ø. The species of ø is the integer ek, where k is the number of connected components in the set Fix(ø) of fixed points of ø and ε = -1 if X \ Fix(ø) is connected and ε = 1 otherwise. A compact Riemann surface X of genus g > 1 is said to be p-hyperelliptic if it admits a conformal involution p, called a p-hyperelliptic involution, for which X/p is an orbifold of genus p. Symmetries of p-hyperelliptic Riemann surfaces has been studied by Klein for p = 0 and by Bujalance and Costa for p > 0. Here we study the species of symmetries of so called pq-hyperelliptic surface defined as a Riemann surface which is p- and q-hyperelliptic simultaneously.References
[1] E. Bujalance, J. Etayo, J. Gamboa, G. Gromadzki: ”Automorphisms Groups of Compact Bordered Klein Surfaces. A Combinatorial Approach”, Lecture Notes in Math. vol. 1439, Springer-Verlag (1990).
[2] E. Bujalance, A.F.Costa: ”On symmetries of p-hyperelliptic Riemann surfaces”, Springer-Verlag (1997),
[3] H. M. Farkas, I. Kra: ”Riemann Surfaces”, Graduate Text in Mathematics, Springer-Verlag (1980)
[4] A.Harnack: ”Uber die Vieltheiligkeit der ebenen algebraischen Kurven”, Math. Ann. 10, (1876), pp. 189-199.
[5] F.Klein: Uber Realitätsverhältnisse bei einem beliebigen Geschlechte zugehörigen Normalkurve der φ” Math. Ann. 42 (1893) 1-29.
[6] A. M. Macbeath: "Action of automorphisms of a compact Riemann surface on the first homology group". Bull. London Math. Soc. 5 (1973), 103-108.
[7] E. Tyszkowska, On pq-hyperelliptic Riemann surfaces, Coll. Math. 103 (1), (2005), 115-120.
[8] E. Tyszkowska, On p-hyperelliptic involutions of Riemann surfaces, Beiträge zur Algebra und Geometrie, to appear.
[2] E. Bujalance, A.F.Costa: ”On symmetries of p-hyperelliptic Riemann surfaces”, Springer-Verlag (1997),
[3] H. M. Farkas, I. Kra: ”Riemann Surfaces”, Graduate Text in Mathematics, Springer-Verlag (1980)
[4] A.Harnack: ”Uber die Vieltheiligkeit der ebenen algebraischen Kurven”, Math. Ann. 10, (1876), pp. 189-199.
[5] F.Klein: Uber Realitätsverhältnisse bei einem beliebigen Geschlechte zugehörigen Normalkurve der φ” Math. Ann. 42 (1893) 1-29.
[6] A. M. Macbeath: "Action of automorphisms of a compact Riemann surface on the first homology group". Bull. London Math. Soc. 5 (1973), 103-108.
[7] E. Tyszkowska, On pq-hyperelliptic Riemann surfaces, Coll. Math. 103 (1), (2005), 115-120.
[8] E. Tyszkowska, On p-hyperelliptic involutions of Riemann surfaces, Beiträge zur Algebra und Geometrie, to appear.
Published
2017-05-08
How to Cite
[1]
E. Tyszkowska, “On symmetries of pq-hyperelliptic Riemann surfaces”, Proyecciones (Antofagasta, On line), vol. 25, no. 2, pp. 179-189, May 2017.
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