Symmetries of Accola-MacLachlan and Kulkarni surfaces. (English) Zbl 0921.14019

Broughton, S. A.; Bujalance, E.; Costa, A. F.; Gamboa, J. M.; Gromadzki, G.

Proc. Am. Math. Soc. 127, No. 3, 637-646 (1999).

Summary: For all \(g \geq 2\) there is a Riemann surface of genus \(g\) whose automorphism group has order \(8g+8\), establishing a lower bound for the possible orders of automorphism groups of Riemann surfaces. Accola and MacLachlan established the existence of such surfaces; we shall call them Accola-MacLachlan surfaces. Later Kulkarni proved that for sufficiently large \(g\) the Accola-MacLachlan surface is unique for \(g= 0,1,2\bmod 4\) and produced exactly one additional surface (the Kulkarni surface) for \(g= 3\bmod 4\).
In this paper we determine the symmetries of these special surfaces, computing the number of ovals and the separability of the symmetries. The results are then applied to classify the real forms of these complex algebraic curves. Explicit equations of these real forms of Accola-MacLachlan surfaces are given in all but one case.

MSC:

14H55	Riemann surfaces; Weierstrass points; gap sequences
14E07	Birational automorphisms, Cremona group and generalizations
14H30	Coverings of curves, fundamental group
14H37	Automorphisms of curves
14N10	Enumerative problems (combinatorial problems) in algebraic geometry

Keywords:

orders of automorphism groups of Riemann surfaces; Kulkarni surface; number of ovals; symmetries; real forms of Accola-MacLachlan surfaces