Multiple prime covers of the riemann sphere

A compact Riemann surface X of genus g≥2 which admits a cyclic group of automorphisms C _q of prime order q such that X/C _q has genus 0 is called a cyclic q-gonal surface. If a q-gonal surface X is also p-gonal for some prime p≠q, then X is called a multiple prime surface. In this paper, we classify all multiple prime surfaces. A consequence of this classification is a proof of the fact that a cyclic q-gonal surface can be cyclic p-gonal for at most one other prime p.     

The gonality of Riemann surfaces under projections by normal coverings

A compact Riemann surface X of genus g≥2 which can be realized as a q-fold, normal covering of a compact Riemann surface of genus p is said to be (q,p)-gonal. In particular the notion of (2,p)-gonality coincides with p-hyperellipticity and (q,0)-gonality coincides with ordinary q-gonality. Here we completely determine the relationship between the gonalities of X and Y for an N-fold normal covering X→Y between compact Riemann surfaces X and Y. As a consequence we obtain classical results due to Maclachlan (1971) [5] and Martens (1977) [6].     

On Singerman symmetries of a class of Belyi Riemann surfaces

By virtue of the Belyi Theorem an algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces and an important class of them consists of Riemann surfaces having the so-called large group of automorphisms. Necessary and sufficient algebraic conditions for these surfaces to be symmetric were found by Singerman in the middle of the seventies and, by a recent result of Köck and Singerman, the algebraic numbers above can be chosen to be real if and only if the respective surface is symmetric. The aim of this paper is to give, in similar terms, the formulas for the number of ovals of the corresponding symmetries, which we refer to as the Singerman symmetries.     

Prime and composite Laurent polynomials

In the paper [J. Ritt, Prime and composite polynomials, Trans. Amer. Math. Soc. 23 (1922) 51-66] Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly polynomial solutions of the functional equation f(p(z))=g(q(z)). In this paper we study the equation above in the case where f,g,p,q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.     

Aut(X, σ) and Aut(X/σ) are not isomorphic in the category of surfaces with nodes

A symmetric Riemann surface is a pair (X,?σ) where X is a Riemann surface and?σ?is an anticonformal involution. We denote by Aut(X,?σ) the subgroup of Aut(X) defined by the automorphisms commuting with σ. There is a natural isomorphism between Aut(X,?σ) and Aut(X/σ). In this article we shall show that this isomorphism does not stand if X is a Riemann surface with nodes.     

Exceptional points in the elliptic-hyperelliptic locus

An exceptional point in the moduli space of compact Riemann surfaces is a unique surface class whose full automorphism group acts with a triangular signature. A surface admitting a conformal involution with quotient an elliptic curve is called elliptic-hyperelliptic; one admitting an anticonformal involution is called symmetric. In this paper, we determine, up to topological conjugacy, the full group of conformal and anticonformal automorphisms of a symmetric exceptional point in the elliptic-hyperelliptic locus. We determine the number of ovals of any symmetry of such a surface. We show that while the elliptic-hyperelliptic locus can contain an arbitrarily large number of exceptional points, no more than four are symmetric.     

On Projecting Symmetries by Unbranched Regular Coverings of Riemann Surfaces

A symmetry of a Riemann surface X of genus g is an antiholomorphic involution σ of X. It is a classical result of Harnack that the set of fixed points of σ consists of k closed Jordan curves, called ovals, for some k, 0 ≤ k ≤ g + 1; when k = g or k = g+1 we say, following Natanzon [8], that σ is an (M – 1)- or an M-symmetry, respectively. Given a Riemann surface X with an M-symmetry, a Riemann surface Y and a regular covering p: X → Y, we prove that Y admits either an M- or an (M – 1)-symmetry and whenever p is unbranched we describe the groups of covering transformations of p. In the case that X is hyperelliptic we calculate as well the number of unbranched regular coverings p: X → Y in which X has an M-symmetry. The first two authors are supported by MTM2005-01637, the third by SAB2005-0049.     

Harmonic maps and asymptotic Teichmüller space

In this paper, the asymptotic boundary behavior of a Hopf differential or the Beltrami coefficient of a harmonic map is investigated and certain compact properties of harmonic maps are established. It is shown that, if f is a quasiconformal harmonic diffeomorphism between two Riemann surfaces and is homotopic to an asymptotically conformal map modulo boundary, then f is asymptotically conformal itself. In addition, we prove that the harmonic embedding map from the Bers space B Q D (X) of an arbitrary hyperbolic Riemann surface X to the Teichmüller space T (X) induces an embedding map from the asymptotic Bers space A B Q D (X), a quotient space of B Q D (X), into the asymptotic Teichmüller space AT (X). The work was supported by a Foundation for the Author of National Excellent Doctoral Dissertation (Grant No. 200518) of PR China and the National Natural Science Foundation of China (Grant No. 10401036).     

On gonality of Riemann surfaces

A compact Riemann surface X is called a (p, n)-gonal surface if there exists a group of automorphisms C of X (called a (p, n)-gonal group) of prime order p such that the orbit space X/C has genus n. We derive some basic properties of (p, n)-gonal surfaces considered as generalizations of hyperelliptic surfaces and also examine certain properties which do not generalize. In particular, we find a condition which guarantees all (p, n)-gonal groups are conjugate in the full automorphism group of a (p, n)-gonal surface, and we find an upper bound for the size of the corresponding conjugacy class. Furthermore we give an upper bound for the number of conjugacy classes of (p, n)-gonal groups of a (p, n)-gonal surface in the general case. We finish by analyzing certain properties of quasiplatonic (p, n)-gonal surfaces. An open problem and two conjectures are formulated in the paper.     

The full automorphism groups of hyperelliptic Riemann surfaces

For every integer g≥2 we obtain the complete list of groups acting as the full automorphisms groups on hyperelliptic Riemann surfaces of genus g. Partially supported by DGICYT PB 89-201 and Science Plan 910021 Partially supported by DGICYT PB 89/379/C02/01 and Science Plan 910021 Partially supported by DGICYT After the typing of this paper we have heard about a Ph.D. Thesis by Britta Krapp on questions related to the problem studied here.     

How many quasiplatonic surfaces?

We show that the number of isomorphism classes of quasiplatonic Riemann surfaces of genus ≦ g has a growth of type The number of non-isomorphic regular dessins of genus ≦ g has the same growth type. Received: 8 February 2005; revised: 2 May 2005     

On <Emphasis Type="Italic">pq</Emphasis>-hyperelliptic Klein surfaces

In this paper we study compact Klein surfaces of algebraic genus d > 1 admitting p- and q-hyperelliptic involutions by which we mean involutions with the orbit spaces having algebraic genera p and q. We give necessary and sufficient conditions for p, q and d to exist such surfaces. It turns out that these conditions are also sufficient for the existence of such surfaces with commuting involutions what allow us to study this class also. We study the spectrum of hyperellipticity degrees of the product of these involutions and topological type of these surfaces. G. Gromadzki was supported by the grant SAB 2005-0049 of the Spanish Ministry of Education and Sciences. E. Tyszkowska was supported by BW 5100-5-0198-6.     

旗传递5-(v,k,2)设计的分类

本文研究了5-(v,k,2)设计的分类问题.利用典型群PSL(2,q)的子群作用于投影线的轨道定理,证明了旗传递5-(v,k,2)设计的自同构群的基柱不能与PSL(2,3n)同构.从而证明了不存在旗传递的5-(v,k,2)设计.     

The topology of balls and Gromov hyperbolicity of Riemann surfaces

We prove that every ball in any non-exceptional Riemann surface with radius less or equal than is either simply or doubly connected. We use this theorem in order to study the hyperbolicity in the Gromov sense of Riemann surfaces. The results clarify the role of punctures and funnels of a Riemann surface in its hyperbolicity.     

Realisierungen des idealen Randes einer Riemannschen Fläche unter konformen Abschließungen

Let a noncompact Riemann surface R of positive finite genus g be given. If f : R → R′ is a conformal mapping of R into a compact Riemann surface R′ of genus g, we have a realization of the ideal boundary of R on the surface R′. We consider (for the fixed R) all the possible R′ and the associated conformal mappings, and study how large the realized boundary can be. To this aim we pass to the (common) universal space ℂ ^g of the Jacobi variety of any R′ and show that the image sets of the ideal boundary of R in ℂ ^g are uniformly bounded.

     

Transcendence degree of zero-cycles and the structure of Chow motives

In the paper we introduce a transcendence degree of a zero-cycle on a smooth projective variety X and relate it to the structure of the motive of X. In particular, we show that in order to prove Bloch’s conjecture for a smooth projective complex surface X of general type with p _g = 0 it suffices to prove that one single point of a transcendence degree 2 in X(ℂ), over the minimal subfield of definition k ⊂ ℂ of X, is rationally equivalent to another single point of a transcendence degree zero over k. This can be of particular interest in the context of Bloch’s conjecture for those surfaces which admit a concrete presentation, such as Mumford’s fake surface, see [Mumford D., An algebraic surface with K ample, (K ²) = 9, p _g = q = 0, Amer. J. Math., 1979, 101(1), 233–244].     

On cyclic <Emphasis Type="Italic">p</Emphasis>-gonal Riemann surfaces with several <Emphasis Type="Italic">p</Emphasis>-gonal morphisms

Let p be a prime number, p > 2. A closed Riemann surface which can be realized as a p-sheeted covering of the Riemann sphere is called p-gonal, and such a covering is called a p-gonal morphism. If the p-gonal morphism is a cyclic regular covering, the Riemann surface is called a cyclic p-gonal Riemann surface. Accola showed that if the genus is greater than (p − 1)² the p-gonal morphism is unique. Using the characterization of p-gonality by means of Fuchsian groups we show that there exists a uniparametric family of cyclic p-gonal Riemann surfaces of genus (p − 1)² which admit two p-gonal morphisms. In this work we show that these uniparametric families are connected spaces and that each of them is the Riemann sphere without three points. We study the Hurwitz space of pairs (X, f), where X is a Riemann surface in one of the above families and f is a p-gonal morphism, and we obtain that each of these Hurwitz spaces is a Riemann sphere without four points.