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On the Branch Loci of Moduli Spaces of Riemann Surfaces of Low Genera
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

Compact Riemann surfaces of genus greater than 1 can be realized as quotient spaces of the hyperbolic plane by the action of Fuchsian groups. The Teichmüller space is the set of all complex structures of Riemann surfaces and the moduli space the set of conformal equivalence classes of Riemann surfaces. For genus greater than two the branch locus of the covering of the moduli space by the Teichmüller space can be identified wi the set of Riemann surfaces admitting non-trivial automorphisms. Here we give the orbifold structure of the branch locus of surfaces of genus 5 by studying the equisymmetric stratification of the branch locus. This gives the orbifold structure of the moduli space.

We also show that the strata corresponding to surfaces with automorphisms of order 2 and 3 belong to the same connected component for every genus. Further we show that the branch locus is connected with the exception of one isolated point for genera 5 and 6, it is connected for genus 7 and it is connected with the exception of two isolated points for genus 8.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press , 2009. , p. 48
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1413
National Category
Computational Mathematics
Identifiers
URN: urn:nbn:se:liu:diva-51519Local ID: LIU-TEK-LIC-2009:21ISBN: 978-91-7393-532-6 (print)OAI: oai:DiVA.org:liu-51519DiVA, id: diva2:275389
Presentation
2009-11-23, Glashuset., Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Supervisors
Available from: 2009-11-05 Created: 2009-11-05 Last updated: 2018-05-30Bibliographically approved
List of papers
1. On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus
Open this publication in new window or tab >>On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus
2012 (English)In: Proceedings of the American Mathematical Society, ISSN 0002-9939, E-ISSN 1088-6826, Vol. 140, no 1, p. 35-45Article in journal (Refereed) Published
Abstract [en]

Let be an integer and let , where denotes the moduli space of compact Riemann surfaces of genus . Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 and 3 belong to the same connected component. We also prove the connectedness of for and with the exception of the isolated points given by Kulkarni.

Place, publisher, year, edition, pages
American Mathematical Society, 2012
Keywords
Moduli spaces, Teichmüller modular group, automorphism group
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-73196 (URN)10.1090/S0002-9939-2011-10881-5 (DOI)000299596000004 ()
Funder
Swedish Research Council, 621-2007-6240
Available from: 2011-12-21 Created: 2011-12-21 Last updated: 2018-09-01

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Bartolini, Gabriel

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