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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. , 48 p.
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: 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: 2015-03-09Bibliographically approved
List of papers
1. ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF THE MODULI SPACE OF RIEMANN SURFACES OF GENUS 4
Open this publication in new window or tab >>ON THE CONNECTEDNESS OF THE BRANCH LOCUS OF THE MODULI SPACE OF RIEMANN SURFACES OF GENUS 4
2010 (English)In: Glasgow Mathematical Journal, ISSN 0017-0895, E-ISSN 1469-509X, Vol. 52, 401-408 p.Article in journal (Refereed) Published
Abstract [en]

Let g be an integer ≥ 3 and let θg = {X ∈ Mg|Aut(X) ≠ 1d}, where Mg denotes the moduli space of a compact Riemann surface. 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 belongs to the same connected component. We also prove the connectedness of θg for g = 5, 6, 7 and 8 with the exception of the isolated points given by Kulkarni.

National Category
Computational Mathematics
Identifiers
urn:nbn:se:liu:diva-51518 (URN)10.1017/S0017089510000091 (DOI)000277348200016 ()
Note

Original Publication: Gabriel Bartolini and Milagros Izquierdo, On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus, 2010, Glasgow Mathematical Journal, (52), 401-408. http://dx.doi.org/10.1017/S0017089510000091 Copyright: Cambridge University Press http://www.cambridge.org/uk/

Available from: 2009-11-05 Created: 2009-11-05 Last updated: 2017-12-12Bibliographically approved

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