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On the Branch Loci of Moduli Spaces of Riemann Surfaces
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2012 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The spaces of conformally equivalent Riemann surfaces, Mg where g ≥ 1, are not manifolds. However the spaces of the weaker Teichmüller equivalence, Tg are known to be manifolds. The Teichmüller space Tg is the universal covering of Mg and Mg is the quotient space by the action of the modular group. This gives Mg an orbifold structure with a branch locus Bg. The branch loci Bg can be identified with Riemann surfaces admitting non-trivial automorphisms for surfaces of genus g ≥ 3. In this thesis we consider the topological structure of Bg. We study the connectedness of the branch loci in general by considering families of isolated strata and we we establish that connectedness is a phenomenon for low genera. Further, we give the orbifold structure of the branch locus of surfaces of genus 4 and genus 5 in particular, by studying the equisymmetric stratification of the branch locus.

Paper 1. In this paper we show that the strata corresponding to actions of order 2 and 3 belong to the same connected component for arbitrary genera. 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.

Paper 2. This paper contains a collection of results regarding components of the branch loci, some of them proved in detail in other papers. It is shown that for any integer d if p is a prime such that p > (d + 2)2, there there exist isolated strata of dimension d in the moduli space of Riemann surfaces of genus (d + 1)(p − 1)/2. It is also shown that if we consider Riemann surfaces as Klein surfaces, the branch loci are connected for every genera due to reflections.

Paper 3. Here we consider surfaces of genus 4 and 5. Here we study the automorphism groups of Riemann surfaces of genus 4 and 5 up to topological equivalence and determine the complete structure of the equisymmetric stratification of the branch locus.

Paper 4. In this paper we establish that the connectedness of the branch loci is a phenomenon for low genera. More precisely we prove that the only genera g where Bg is connected are g = 3, 4, 13, 17, 19, 59.

Place, publisher, year, edition, pages
Linköping: Linköping University Electronic Press, 2012. , 45 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1440
National Category
Geometry
Identifiers
URN: urn:nbn:se:liu:diva-77449ISBN: 978-91-7519-913-9 (print)OAI: oai:DiVA.org:liu-77449DiVA: diva2:527079
Public defence
2012-06-05, Planck, Fysikhuset, ingång 57, Campus Valla, Linköpings universitet, Linköping, 10:15 (English)
Opponent
Supervisors
Available from: 2012-06-05 Created: 2012-05-16 Last updated: 2015-03-09Bibliographically approved
List of papers
1. On the connectedness of the branch locus of the moduli space of Riemann surfaces
Open this publication in new window or tab >>On the connectedness of the branch locus of the moduli space of Riemann surfaces
2010 (English)In: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, ISSN 1578-7303, Vol. 104, no 1, 81-86 p.Article in journal (Refereed) Published
Abstract [en]

The moduli space M-g of compact Riemann surfaces of genus g has the structure of an orbifold and the set of singular points of such orbifold is the branch locus B-g. In this article we present some results related with the topology of B-g. We study the connectedness of B-g for g andlt;= 8, the existence of isolated equisymmetric strata in the branch loci and finally we stablish the connectedness of the branch locus of the moduli space of Riemann surfaces considered as Klein surfaces. We just sketch the proof of some of the results; complete proofs will be published elsewhere.

Place, publisher, year, edition, pages
REAL ACAD CIENCIAS EXACTAS FISICAS and NATURALES, CALLE VALVERDE 22, MADRID, 28004, SPAIN, 2010
Keyword
Riemann surface, moduli space, automorphism
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-54848 (URN)10.5052/RACSAM.2010.08 (DOI)000276304700008 ()
Note
Original Publication: Gabriel Bartolini, Antonio F Costa, Milagros Izquierdo and Ana M Porto, On the connectedness of the branch locus of the moduli space of Riemann surfaces, 2010, REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, (104), 1, 81-86. http://dx.doi.org/10.5052/RACSAM.2010.08 Copyright: Real Academia de Ciencias, Espana Available from: 2010-04-16 Created: 2010-04-16 Last updated: 2015-03-09
2. 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, 35-45 p.Article 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
Keyword
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: 2017-12-08
3. On the Orbifold Structure of the Moduli Space of Riemann Surfaces of Genera Four and Five
Open this publication in new window or tab >>On the Orbifold Structure of the Moduli Space of Riemann Surfaces of Genera Four and Five
2014 (English)In: REVISTA DE LA REAL ACADEMIA DE CIENCIAS EXACTAS FISICAS Y NATURALES SERIE A-MATEMATICAS, ISSN 1578-7303, Vol. 108, no 2, 769-793 p.Article in journal (Refereed) Published
Abstract [en]

The moduli space Mg, of compact Riemann surfaces of genus g has orbifold structure since Mg is the quotient space of the Tiechmüller space by the action of the mapping class group. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space we find the orbifold structure of the moduli spaces of Riemann surfaces of genera 4 and 5.

Place, publisher, year, edition, pages
Springer, 2014
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-78016 (URN)10.1007/s13398-013-0140-8 (DOI)000340875100032 ()
Available from: 2012-06-04 Created: 2012-06-04 Last updated: 2017-04-10Bibliographically approved
4. On the connected branch loci of moduli spaces
Open this publication in new window or tab >>On the connected branch loci of moduli spaces
(English)Manuscript (preprint) (Other academic)
Abstract [en]

The moduli space Mg of compact Riemann surfaces of genus g has orbifold structure and the set of singular points of such orbifold is the branch locus Bg. In this article we show that Bg is connected exactly for genera three, four, thirteen, seventeen, nineteen and fitfynine by the use automorphisms of order 5 and 7 of Riemann surfaces, and calculations with GAP for some small genera.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-78018 (URN)
Available from: 2012-06-04 Created: 2012-06-04 Last updated: 2012-06-05Bibliographically approved

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