On the connectedness of the locus of real Riemann surfaces
2002 (English)In: Annales Academiae Scientiarum Fennicae Mathematica, ISSN 1239-629X, Vol. 27, no 2, 341-356 p.Article in journal (Refereed) Published
It is well known that the functorial equivalence between pairs (X, sigma), where X is a Riemann surface which admits an antiholomorphic involution (symmetry) sigma: X --> X, and real algebraic curves. We shall refer to such Riemann surfaces as real Riemann surfaces, following Klein's terminology. We consider the sets M-g(R) and M-g(2R) of real curves and :real hyperelliptic curves, respectively in the moduli space M-g of complex algebraic curves of genus g. In this paper we prove that any real hyperelliptic Riemann surface can be quasiconformally deformed, preserving the real and hyperelliptic character, to a real hyperelliptic Riemann surface (X, sigma), such that X admits a symmetry tau, where Fix (tau) is connected and non-separating. As a consequence, we obtain the connectedness of the sets M-g(2R)(subset of M-g) of all real hyperelliptic Riemann surfaces of genus g and M-g(R)(subset of M-g) of all real Riemann surfaces of given genus g using a procedure different from the one given by Seppala for M-g(2R) and Buser, Seppala and Silhol for M-g(R). A Riemann surface X is called a p-gonal Riemann surface, where p is a prime, if there exists a p-fold covering map from X onto the Riemann sphere. We prove ill this paper that the subset of real p-gonal Riemann surfaces, p greater than or equal to 3, is not a connected subset of M-g in general. This generalizes a result of Gross and Harris for real trigonal algebraic curves.
Place, publisher, year, edition, pages
2002. Vol. 27, no 2, 341-356 p.
National CategoryEngineering and Technology
IdentifiersURN: urn:nbn:se:liu:diva-48760OAI: oai:DiVA.org:liu-48760DiVA: diva2:269656