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Mirrors on Riemann surfaces. (English) Zbl 0851.30025
Bokut’, L. A. (ed.) et al., Second international conference on algebra dedicated to the memory of A. I. Shirshov. Proceedings of the conference on algebra, August 20-25, 1991, Barnaul, Russia. Providence, RI: American Mathematical Society. Contemp. Math. 184, 411-417 (1995).
Let \(X\) be a compact Riemann surface of genus \(g > 1\). A symmetry \(S\) of \(X\) is an anticonformal involution. The fixed point set of \(S\) consists of \(k \leq g + 1\) disjoint closed curves. Each such closed curve is called a mirror of \(S\). Let \(G\) be a group of conformal and anticonformal automorphisms of \(X\). Let \(S_1\), \(S_2, \dots\) denote the representatives of conjugacy classes of symmetries in \(G\). In this paper the author studies all the possible numbers \(|S_1 |\), \(|S_2 |, \dots\) (where \(|S_i |\) denotes the numbers of mirrors of \(S_i)\) in two important cases: (i) if \(G\) is the dihedral group of order \(2n\), where he obtains a similar result to that obtained by S. M. Natanzon [Tr. Mosk. Mat. O.-va 51, 3-53 (1988; Zbl 0692.14020)] but by different methods. (ii) if \(G \approx Z_2^m\) \((m \geq 3)\) (the case where the symmetries commute), in this case the author obtains that \(|S_1 |+ \cdots + |S_m |\leq 2g + 2^{m - 3} (9 - m) - 2\).
For the entire collection see [Zbl 0824.00029].

30F10 Compact Riemann surfaces and uniformization
14H25 Arithmetic ground fields for curves