On gonality of Riemann surfaces
type: Item
instance of: scholarly article
title: On gonality of Riemann surfaces
zbMATH Open document ID: 1213.14068
author: G. Gromadzki
author: Anthony Weaver
author: Aaron Wootton
published in: Geometriae Dedicata
publication date: 2010-11-23T00:00:00Z
review text: 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. T…
Mathematics Subject Classification ID: 14J10
Mathematics Subject Classification ID: 14J50
Mathematics Subject Classification ID: 30F10
Mathematics Subject Classification ID: 30F20
zbMATH DE Number: 5818354
zbMATH Keywords: automorphism groups of Riemann surfaces
zbMATH Keywords: hyperelliptic Riemann surfaces
zbMATH Keywords: \(p\)-hyperelliptic Riemann surfaces
zbMATH Keywords: \(p\)-gonal Riemann surfaces
zbMATH Keywords: Fuchsian groups
zbMATH Keywords: uniformization theorem
reviewed by: Rubén A. Hidalgo
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: s10711-010-9459-x
OpenAlex ID: W2142190866
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DOI: 10.1007/S10711-010-9459-X
description: scientific article; zbMATH DE number 5818354