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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 \(g=8m^2+1, m\geq 2\).

MSC:
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14H55 Riemann surfaces; Weierstrass points; gap sequences
30F10 Compact Riemann surfaces and uniformization
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