On gonality of Riemann surfaces
A closed Riemann surface \(S\), say of genus \(g\), is called \((p,n)\)-gonal if \(S\) has a conformal automorphism \(\phi\) of order \(p\) and so that the quotient orbifold \(S/\langle \phi \rangle\) has genus \(n\). In this paper the authors assume \(p\) to be a prime. In this case, the cyclic group \(\langle \phi \rangle\) is called a \((p,n)\)-gonal group of \(S\). The main result of the paper is Theorem 4.5., which states that if \(g \geq 2\) and \(p>2n+1\), then all \((p,n)\)-gonal groups are conjugate in the full group of conformal automorphisms of the surface. The case \(n=0\) was previously obtained by \textit{G. González-Diez} [Ann. Mat. Pura Appl., IV. Ser. 168, 1--15 (1995; Zbl 0846.30029)]. If \(g>2pn+(p-1)^{2}\), then there is at most one \((p,n)\)-gonal group as consequence of the Castelnuovo-Severi theorem. Theorem 4.10., states that for any fixed prime \(p\) and any integer \(k \geq 2\), there exists a \((p,n)\)-gonal surface, for some \(n>1\), containing exactly \(k\) non-conjugate \((p,n)\)-gonal groups (acting with fixed points). The proof is given by explicit constructions. Theorem 4.11., states an upper bound for the number of conjugacy classes of \((p,n)\)-gonal groups (acting with fixed points) that a closed Riemann surface of genus \(g \geq 2\) may have. The upper bound does not carry the integer \(n\). In particular, such an upper bound should be to generous. It is asked if a better upper bound can be obtained, one that should necessarily contain the integer \(n\). Theorem 5.2., states the maximum number of conjugate \((p,n)\)-gonal groups (acting with fixed points) that the surface may have for \(p>2n+1\). The final section of the paper concerns quasiplatonic Riemann surfaces (that is, the orbifold obtained by quotient the surface by its full group of conformal automorphism is the Riemann sphere with exactly three cone points) which are \((p,n)\)-gonal. First, the author provide upper bounds for the full group of conformal automorphisms. Then, they state Theorem 6.2., which asserts that, for \(n>1\), there are only finitely many quasiplatonic surfaces admitting a \((p,n)\)-gonal group (acting with fixed points). In fact, if \(g>42p(n-1)\), then there is not such situation. The paper is nicely written and easy to follow by non-specialist.
scientific article; zbMATH DE number 5818354
scientific article