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Real parts of complex algebraic curves. (English) Zbl 0736.30031
Real analytic and algebraic geometry, Proc. Conf., Trento/Italy 1988, Lect. Notes Math. 1420, 81-110 (1990).
[For the entire collection see Zbl 0686.00007.]
The authors study compact Riemann surfaces of genus \(g\geq 3\) whose group \(A(S)\) of analytic automorphisms is \({\mathbb{Z}}_ p\) and which admit an antianalytic involution \(T\). In particular, they determine for which odd primes \(p\), \(m>0\), \(g\geq 3\) and \(k\) does there exist such a surface \(S\) so that the quotient \(S/A(S)\) has genus \(m\) and so that \(S/T\) has \(k\) boundary components (for both \(S/T\) orientable and not). The relation with real parts of complex algebraic curves occurs since one may take \(T\) to be complex conjugation if the curve is defined over the reals, then \(k\) is the number of components of the real points.

MSC:
30F10 Compact Riemann surfaces and uniformization
14H55 Riemann surfaces; Weierstrass points; gap sequences
14P25 Topology of real algebraic varieties
30F35 Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization)
20H15 Other geometric groups, including crystallographic groups
14H99 Curves in algebraic geometry