Fixed points of automorphisms of real algebraic curves (Q2476185)

A real algebraic curve is a pair \((X, \sigma)\), where \(\sigma\) is an antianalytic involution of the compact Riemann surface \(X\). The real points of \((X,\sigma)\) are those points of \(X\) fixed by \(\sigma\). Let \(k_X\) denote the number of connected components of the set \(\text{Fix}(\sigma)\) of real points of \((X, \sigma)\), and let \(g\geq 2\) be the genus of \((X, \sigma),\) defined as the genus of \(X\). The automorphisms of \((X, \sigma)\) are those analytic or antianalytic automorphisms \(f\) of \(X\) such that \(f\circ\sigma=\sigma\circ f\). Let \(\mu_{\mathbb R}(f)\) denote the number of real points of \(X\) fixed by \(f\). The paper under review presents, under some assumptions on the set of real points of \((X, \sigma)\), relations between the integers \(g\), \(k_X\), \(\mu_{\mathbb R}(f)\) and the order \(| f| \) of \(f\). In most cases such relations are upper bounds of \(\mu_{\mathbb R}(f)\), and I do not see obvious its sharpness. As a byproduct, some consequences about the maximum order of either cyclic or abelian automorphisms groups of \((X, \sigma)\) are derived. The author also obtains an upper bound for the number of automorphisms of a real hyperelliptic curve. As the author explains in the introduction of the article, some of the results it contains are well known some years ago. Although the credits are not properly attributed in the paper, I should mention that the upper bounds for the abelian automorphisms groups of real algebraic curves are known since 1990, as were independently obtained by \textit{S. Natanzon} [Russ. Math. Surv. 45, No. 6, 53--108 (1990); translation from Usp. Mat. Nauk 45, No. 6(276), 47--90 (1990; Zbl 0734.30037)] and \textit{E. Bujalance, J. J. Etayo, J. M. Gamboa} and \textit{G. Gromadzki} [``Automorphism groups of compact bordered Klein surfaces. A combinatorial approach.'' Lect. Notes Math. 1439. (Berlin) etc.: Springer-Verlag. (1990; Zbl 0709.14021)]. Also the upper bounds for the automorphism groups of real hyperelliptic curves are well known, at least after the work by \textit{E. Bujalance, F. J. Cirre, J. M. Gamboa} and \textit{G. Gromadzki} [``Symmetry types of hyperelliptic Riemann surfaces.'' Mém. Soc. Math. Fr,. Nouv. Sér. 86, (2001; Zbl 1078.14044)]. Reviewer's remark: Probably the author does not know the paper of \textit{C. Corrales, J. M. Gamboa} and \textit{G. Gromadzki} [Glasg. Math. J. 41, 183--189 (1999; Zbl 0927.30027)]. It is devoted to find sharp upper bounds for the order of an automorphism of a real algebraic curve with real points as a function of the genus of the curve, its number of connected components and the number of points fixed by the automorphism. Although the language used in our paper and the paper under review are rather different, the mathematical arguments are quite similar: covering theory and, in particular, Riemann-Hurwitz formula! Unfortunately, I have been unable to obtain the results of one paper from the ones in the other (in both directions).