scholarly journals ON THE CONNECTEDNESS OF THE BRANCH LOCI OF NON-ORIENTABLE UNBORDERED KLEIN SURFACES OF LOW GENUS

2014 ◽  
Vol 57 (1) ◽  
pp. 211-230 ◽  
Author(s):  
E. BUJALANCE ◽  
J. J. ETAYO ◽  
E. MARTÍNEZ ◽  
B. SZEPIETOWSKI
Keyword(s):  
Moduli Space ◽  
Prime Order ◽  
Topological Conjugacy ◽  
Branch Locus ◽  
Klein Surfaces ◽  
Topological Genus

AbstractThis paper is devoted to determine the connectedness of the branch loci of the moduli space of non-orientable unbordered Klein surfaces. We obtain a result similar to Nielsen's in order to determine topological conjugacy of automorphisms of prime order on such surfaces. Using this result we prove that the branch locus is connected for surfaces of topological genus 4 and 5.

scholarly journals On the connectedness of the branch locus of moduli space of hyperelliptic Klein surfaces with one boundary

2017 ◽  
Vol 28 (05) ◽  
pp. 1750038 ◽  
Author(s):  
Antonio F. Costa ◽  
Milagros Izquierdo ◽  
Ana Maria Porto
Keyword(s):  
Moduli Space ◽  
Boundary Component ◽  
Branch Locus ◽  
Klein Surfaces ◽  
Genus Formula

In this work, we prove that the hyperelliptic branch locus of orientable Klein surfaces of genus [Formula: see text] with one boundary component is connected and in the case of non-orientable Klein surfaces it has [Formula: see text] components, if [Formula: see text] is odd, and [Formula: see text] components for even [Formula: see text]. We notice that, for non-orientable Klein surfaces with two boundary components, the hyperelliptic branch loci are connected for all genera.

A NOTE ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF RATIONAL MAPS

2017 ◽  
Vol 60 (1) ◽  
pp. 199-207
Author(s):  
RUBEN A. HIDALGO ◽  
SAÚL QUISPE
Keyword(s):  
Moduli Space ◽  
Singular Locus ◽  
Equivalence Classes ◽  
Rational Maps ◽  
Branch Locus ◽  
Cubic Curve ◽  
Dimension 2 ◽  
Holomorphic Automorphisms

AbstractMilnor proved that the moduli space Md of rational maps of degree d ≥ 2 has a complex orbifold structure of dimension 2(d − 1). Let us denote by ${\mathcal S}$d the singular locus of Md and by ${\mathcal B}$d the branch locus, that is, the equivalence classes of rational maps with non-trivial holomorphic automorphisms. Milnor observed that we may identify M2 with ℂ2 and, within that identification, that ${\mathcal B}$2 is a cubic curve; so ${\mathcal B}$2 is connected and ${\mathcal S}$2 = ∅. If d ≥ 3, then it is well known that ${\mathcal S}$d = ${\mathcal B}$d. In this paper, we use simple arguments to prove the connectivity of ${\mathcal S}$d.

scholarly journals ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF THE MODULI SPACE OF RIEMANN SURFACES OF GENUS 4

2010 ◽  
Vol 52 (2) ◽  
pp. 401-408 ◽  
Author(s):  
ANTONIO F. COSTA ◽  
MILAGROS IZQUIERDO
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Fuchsian Groups ◽  
Branch Locus

AbstractUsing uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space of surfaces of genus 4, we prove its connectedness. As a consequence, one can deform a surface of genus 4 with automorphisms, i.e. symmetric, to any other symmetric genus 4 surface through a path consisting entirely of symmetric surfaces.

scholarly journals THE HYPERELLIPTIC MAPPING CLASS GROUP OF KLEIN SURFACES

2001 ◽  
Vol 44 (2) ◽  
pp. 351-363 ◽  
Author(s):  
E. Bujalance ◽  
A. F. Costa ◽  
J. M. Gamboa
Keyword(s):  
Moduli Space ◽  
Mapping Class Group ◽  
Class Group ◽  
Algebraic Curves ◽  
Topological Type ◽  
Mapping Class ◽  
Klein Surfaces ◽  
Primary 14H10 ◽  
Generators And Relations ◽  
Real Algebraic Curves

AbstractIn this paper we study the algebraic structure of the hyperelliptic mapping class group of Klein surfaces, which is closely related to the mapping class group of punctured discs. This group plays an important role in the study of the moduli space of hyperelliptic real algebraic curves. Our main result provides a presentation by generators and relations for the hyperelliptic mapping class group of surfaces of prescribed topological type.AMS 2000 Mathematics subject classification: Primary 14H10; 20H10; 30F50

scholarly journals On the connectedness of the branch locus of the moduli space of Riemann surfaces

2010 ◽  
Vol 104 (1) ◽  
pp. 81-86 ◽  
Author(s):  
Gabriel Bartolini ◽  
Antonio F. Costa ◽  
Milagros Izquierdo ◽  
Ana M. Porto
Keyword(s):  
Moduli Space ◽  
Riemann Surfaces ◽  
Branch Locus

scholarly journals Minimum topological genus of compact bordered Klein surfaces admitting a prime-power automorphism

1995 ◽  
Vol 37 (2) ◽  
pp. 221-232 ◽  
Author(s):  
E. Bujalance ◽  
J. M. Gamboa ◽  
C. Maclachlan
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Riemann Surfaces ◽  
Prime Power ◽  
Compact Riemann Surface ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Abelian Case ◽  
Power Automorphism ◽  
Topological Genus

In the nineteenth century, Hurwitz [8] and Wiman [14] obtained bounds for the order of the automorphism group and the order of each automorphism of an orientable and unbordered compact Klein surface (i. e., a compact Riemann surface) of topological genus g s 2. The corresponding results of bordered surfaces are due to May, [11], [12]. These may be considered as particular cases of the general problem of finding the minimum topological genus of a surface for which a given finite group G is a group of automorphisms. This problem was solved for cyclic and abelian G by Harvey [7] and Maclachlan [10], respectively, in the case of Riemann surfaces and by Bujalance [2], Hall [6] and Gromadzki [5] in the case of non-orientable and unbordered Klein surfaces. In dealing with bordered Klein surfaces, the algebraic genus—i. e., the topological genus of the canonical double covering, (see Alling-Greenleaf [1])—was minimized by Bujalance- Etayo-Gamboa-Martens [3] in the case where G is cyclic and by McCullough [13] in the abelian case.

scholarly journals On the order of automorphism groups of Klein surfaces

1985 ◽  
Vol 26 (1) ◽  
pp. 75-81 ◽  
Author(s):  
J. J. Etayo Gordejuela
Keyword(s):  
General Result ◽  
Special Interest ◽  
Riemann Surfaces ◽  
Prime Order ◽  
Automorphism Groups ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Group Of Automorphisms

A problem of special interest in the study of automorphism groups of surfaces are the bounds of the orders of the groups as a function of the genus of the surface.May has proved that a Klein surface with boundary of algebraic genus p has at most 12(p–1) automorphisms [9].In this paper we study the highest possible prime order for a group of automorphisms of a Klein surface. This problem was solved for Riemann surfaces by Moore in [10]. We shall use his results for studying the Klein surfaces that are not Riemann surfaces. The more general result that we obtain is the following: if X is a Klein surface of algebraic genus p, and G is a group of automorphisms of X, of prime order n, then n ≤ p + 1.

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