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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard


IMPACT FACTOR 2018: 0.789

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1615-7168
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Volume 19, Issue 4

Issues

Limit points of the branch locus of đť“śg

Raquel DĂ­az
  • Corresponding author
  • Departamento de GeometrĂ­a y TopologĂ­a, Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, Madrid, España
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/ Víctor González-Aguilera
Published Online: 2019-06-23 | DOI: https://doi.org/10.1515/advgeom-2018-0029

Abstract

Let 𝓜g be the moduli space of compact connected hyperbolic surfaces of genus g ≥ 2, and 𝓑g ⊂ 𝓜g its branch locus. Let Mg^ be the Deligne–Mumford compactification of the moduli space of smooth, complete, connected surfaces of genus g ≥ 2 over ℂ. The branch locus 𝓑g is stratified by smooth locally closed equisymmetric strata, where a stratum consists of hyperbolic surfaces with equivalent action of their orientation-preserving isometry group. Any stratum can be determined by a certain epimorphism Φ. In this paper, for any of these strata, we describe the topological type of its limits points in 𝓜͡g in terms of Φ. We apply our method to the 2-complex dimensional stratum corresponding to the pyramidal hyperbolic surfaces.

Keywords: Moduli space; stratification; noded Riemann surfaces

MSC 2010: Primary 32G15; Secondary 14H10

References

  • [1]

    W. Abikoff, Degenerating families of Riemann surfaces. Ann. of Math. (2) 105 (1977), 29–44. MR0442293 Zbl 0347.32010CrossrefGoogle Scholar

  • [2]

    J. D. Achter, R. Pries, The integral monodromy of hyperelliptic and trielliptic curves. Math. Ann. 338 (2007), 187–206. MR2295509 Zbl 1129.11027CrossrefWeb of ScienceGoogle Scholar

  • [3]

    T. Ashikaga, M. Ishizaka, Classification of degenerations of curves of genus three via Matsumoto–Montesinos’ theorem. Tohoku Math. J. (2) 54 (2002), 195–226. MR1904949 Zbl 1094.14006CrossrefGoogle Scholar

  • [4]

    G. Bartolini, A. F. Costa, M. Izquierdo, On the connectivity of branch loci of moduli spaces. Ann. Acad. Sci. Fenn. Math. 38 (2013), 245–258. MR3076808 Zbl 1279.14032CrossrefGoogle Scholar

  • [5]

    G. Bartolini, A. F. Costa, M. Izquierdo, On the orbifold structure of the moduli space of Riemann surfaces of genera four and five. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM 108 (2014), 769–793. MR3249974 Zbl 1297.14031Google Scholar

  • [6]

    L. Bers, On spaces of Riemann surfaces with nodes. Bull. Amer. Math. Soc. 80 (1974), 1219–1222. MR0361165 Zbl 0294.32017CrossrefGoogle Scholar

  • [7]

    M. Boileau, S. Maillot, J. Porti, Three-dimensional orbifolds and their geometric structures, volume 15 of Panoramas et Synthèses. Société Mathématique de France, Paris 2003. MR2060653 Zbl 1058.57009Google Scholar

  • [8]

    S. A. Broughton, The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups. Topology Appl. 37 (1990), 101–113. MR1080344 Zbl 0747.32017CrossrefGoogle Scholar

  • [9]

    S. A. Broughton, Classifying finite group actions on surfaces of low genus. J. Pure Appl. Algebra 69 (1991), 233–270. MR1090743 Zbl 0722.57005CrossrefGoogle Scholar

  • [10]

    M. Cornalba, On the locus of curves with automorphisms. Ann. Mat. Pura Appl. (4) 149 (1987), 135–151. MR932781 Zbl 0649.14013CrossrefGoogle Scholar

  • [11]

    A. F. Costa, V. Gonz\’alez-Aguilera, Limits of equisymmetric 1-complex dimensional families of Riemann surfaces. Math. Scand. 121 (2017), 26–48. MR3708962 Zbl 06796586Web of ScienceCrossrefGoogle Scholar

  • [12]

    A. F. Costa, M. Izquierdo, On the connectedness of the branch locus of the moduli space of Riemann surfaces of genus 4. Glasg. Math. J. 52 (2010), 401–408. MR2610983 Zbl 1195.30064CrossrefWeb of ScienceGoogle Scholar

  • [13]

    A. F. Costa, M. Izquierdo, H. Parlier, Connecting p-gonal loci in the compactification of moduli space. Rev. Mat. Complut. 28 (2015), 469–486. MR3344087 Zbl 1317.14062CrossrefWeb of ScienceGoogle Scholar

  • [14]

    P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. Inst. Hautes études Sci. Publ. Math. no. 36 (1969), 75–109. MR0262240 Zbl 0181.48803CrossrefGoogle Scholar

  • [15]

    V. Gonz\’alez-Aguilera, R. E. Rodrí guez, On principally polarized abelian varieties and Riemann surfaces associated to prisms and pyramids. In: Lipa’s legacy (New York, 1995), volume 211 of Contemp. Math., 269–284, Amer. Math. Soc. 1997. MR1476992 Zbl 0926.32019Google Scholar

  • [16]

    W. Harvey, Chabauty spaces of discrete groups. Princeton Univ. Press 1974. MR0364629 Zbl 0313.32028Google Scholar

  • [17]

    J. H. Hubbard, S. Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves. J. Differential Geom. 98 (2014), 261–313. MR3263519 Zbl 1318.32019CrossrefGoogle Scholar

  • [18]

    H. Kimura, Classification of automorphism groups, up to topological equivalence, of compact Riemann surfaces of genus 4. J. Algebra 264 (2003), 26–54. MR1980684 Zbl 1027.30063CrossrefGoogle Scholar

  • [19]

    A. M. Macbeath, D. Singerman, Spaces of subgroups and Teichm\"uller space. Proc. London Math. Soc. (3) 31 (1975), 211–256. MR0397022 Zbl 0314.32012Google Scholar

  • [20]

    Y. Matsumoto, J. M. Montesinos-Amilibia, A proof of Thurston’s uniformization theorem of geometric orbifolds. Tokyo J. Math. 14 (1991), 181–196. MR1108165 Zbl 0732.57015CrossrefGoogle Scholar

  • [21]

    Y. Matsumoto, J. M. Montesinos-Amilibia, Pseudo-periodic maps and degeneration of Riemann surfaces. Springer 2011. MR2839459 Zbl 1239.57001Google Scholar

  • [22]

    R. Miranda, Graph curves and curves on K3 surfaces. In: Lectures on Riemann surfaces (Trieste, 1987), 119–176, World Sci. Publ., Teaneck, NJ 1989. MR1082353 Zbl 0800.14014Google Scholar

About the article

Received: 2017-03-19

Revised: 2017-12-28

Published Online: 2019-06-23

Published in Print: 2019-10-25


Communicated by: J. Ratcliffe

Funding: The first author was partially supported by the Project MTM2012-31973. The second author was partially supported by Project PIA, ACT 1415.


Citation Information: Advances in Geometry, Volume 19, Issue 4, Pages 505–526, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2018-0029.

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