Proyecciones Journal of Mathematics
Vol. 25, No 2, pp. 179-189, August 2006.
Universidad Católica del Norte
Antofagasta - Chile

ON SYMMETRIES OF PQ-HYPERELLIPTIC RIEMANN SURFACES *

EWA TYSZKOWSKA

UNIVERSITY OF GDANSK, POLAND

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.

keywords : p-hyperelliptic Riemann surface, automorphisms of Riemann surface, fixed points of automorphism, symmetry

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: Über 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.

*Supported by BW 5100-5-0089-5

Received : May 2006. Accepted : June 2006

Ewa Tyszkowska
Institute of Mathematics
University of Gdansk
Wita Stwosza 57,
80-952 Gdánsk
Poland
e-mail : Ewa.Tyszkowska@math.univ.gda.pl