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On a Harnack-Natanzon theorem for the family of real forms of Riemann surfaces. (English) Zbl 0885.14026
Let \(X\) be a compact Riemann surface of genus \(g\geq 2\). A symmetry of \(X\) is an antiholomorphic involution \(\sigma ,\) i.e. an orientation-reversing automorphism of \(X\) of order 2. An old theorem of Harnack states that a symmetry of a compact Riemann surface \(X\) of genus \(g\geq 2\) has at most \(g+1\) disjoint simple closed curves of fixed points, each of which is called an oval of \(X\). Much more recently Natanzon proved that for \(\nu (g)\) being the maximum number of ovals that a surface of genus \(g\) admits, \(\nu (g)\leq 42(g-1)\). It is known, that a surface \(X_g\) corresponding to the equation \(y^2=x^{2(g+1)}-1\) admits a symmetry with \(g+1\) ovals and so in particular \(\nu (g)\geq g+1\). It the paper under review it is proved:
Theorem 4.2. It is valid, that \(\nu (g)\leq 12(g-1)\) for \(g\neq 2,3,5,7,9\). For \(g=2,3,5,7\) and 9, \(\nu (g)=24,36,72,126\) and \(100,\) respectively. Moreover, \(\nu (g)=12(g-1)\) for all values of \(g\) of the form \(g=8m^2+1, m\geq 2\).

14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H55 Riemann surfaces; Weierstrass points; gap sequences
30F10 Compact Riemann surfaces and uniformization
Full Text: DOI
[1] Alling, N.L.; Greenleaf, N., Foundations of the theory of Klein surfaces, () · Zbl 0203.23603
[2] Ballico, E., Real moduli of complex objects: surfaces and bundles, Monatshefte math., 115, 13-26, (1993) · Zbl 0790.14045
[3] Bochnak, J.; Coste, M.; Roy, M.F., Géométrie algébrique Réelle, () · Zbl 0633.14016
[4] Broughton, A.; Bujalance, E.; Costa, A.F.; Gamboa, J.M.; Gromadzki, G., Symmetries of Riemann surfaces on which PSL(2, q) acts as Hurwitz automorphisms group, J. pure appl. algebra, 106, 113-126, (1996) · Zbl 0847.30026
[5] Bujalance, E.; Etayo, J.J.; Gamboa, J.M.; Gromadzki, G., Automorphisms groups of compact bordered Klein surfaces, A combinatorial approach, () · Zbl 1213.30074
[6] Bujalance, E.; Gromadzki, G.; Singerman, D., On the number of real curves associated to a complex algebraic curve, (), 507-513 · Zbl 0820.30025
[7] Gamboa, J.M., Compact Klein surfaces with boundary viewed as real compact smooth algebraic curves, Mem. R. acad. ci. Madrid, 27, (1991) · Zbl 0749.14021
[8] Gromadzki, G., Groups of automorphisms of compact Riemann and Klein surfaces, (1993), University Press, WSP Bydgoszcz
[9] Harnack, A., Über die vieltheiligkeit der ebenen algebraischen kurven, Math. ann., 10, 189-199, (1876) · JFM 08.0317.04
[10] Hoare, A.H.M., Subgroups of NEC groups and finite permutation groups, Quart J. math. Oxford (2), 41, 45-59, (1990) · Zbl 0693.20044
[11] Macbeath, A.M., The classification of non-Euclidean crystallographic groups, Can. J. math., 19, 1192-1205, (1967) · Zbl 0183.03402
[12] Natanzon, S.M., Moduli of real algebraic curves, Uspekhi mat. nauk, 30, 251-252, (1975), (in Russian)
[13] Natanzon, S.M.; Natanzon, S.M., Moduli spaces of real curves, Trudy moskov mat. obsch., Trans. Moscow math. soc., 1, 233-272, (1980) · Zbl 0452.14006
[14] Natanzon, S.M.; Natanzon, S.M., On the order of a finite group of homeomorphisms of a surface into itself and the real number of real forms of a complex algebraic curve, Dokl. akad. nauk SSSR, Sov. math. dokl., 19, 1195-1199, (1978) · Zbl 0417.57003
[15] Natanzon, S.M.; Natanzon, S.M., On the total number of ovals of real forms of complex algebraic curves, Uspekhi mat. nauk, Russian math. surveys, 35, 223-224, (1980) · Zbl 0467.14009
[16] Natanzon, S.M.; Natanzon, S.M., Finite groups of homeomorphisms of surfaces and real forms of complex algebraic curves, Trudy moscov mat. obsch, Trans. Moscow math. soc., 51, 1-51, (1989) · Zbl 0692.14020
[17] Natanzon, S.M.; Natanzon, S.M., Klein surfaces, Uspekhi mat. nauk, Russian math. surveys, 45, 43-108, (1990) · Zbl 0734.30037
[18] Preston, R., Projective structures and fundamental domains on compact Klein surfaces, ()
[19] Seppälä, M.; Silhol, R., Moduli spaces for real algebraic curves and real abelian varieties, Math. Z., 201, 151-165, (1989) · Zbl 0645.14012
[20] Silhol, R., Compactifications of moduli in real algebraic geometry, Invent. math., 107, 151-202, (1992) · Zbl 0777.14014
[21] Singerman, D., Non-Euclidean crystallographic groups and Riemann surfaces, () · Zbl 0232.30012
[22] Singerman, D., Symmetries of Riemann surfaces with large automorphism group, Math. ann., 210, 17-32, (1974) · Zbl 0272.30022
[23] Singerman, D., On the structure of non-Euclidean crystallographic groups, (), 233-240 · Zbl 0284.20053
[24] Wilkie, H.C., On non-Euclidean crystallographic groups, Math. Z., 91, 87-102, (1966) · Zbl 0166.02602
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