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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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