Invariant Spin Structures on Riemann Surfaces
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 457-477.

Dans ce travail, nous étudions l’action du groupe d’automorphismes conformes d’une surface de Riemann de genre supérieur à deux sur ses structures spin. Nous caractérisons de telles surfaces qui admettent un automorphisme non-trivial fixant soit toutes les structures spin à la fois, soit seulement une. Les cas des courbes hyperelliptiques et de la quartique de Klein sont analysés en détail.

We investigate the action of the automorphism group of a closed Riemann surface of genus at least two on its set of theta characteristics (or spin structures). We give a characterization of those surfaces admitting a non-trivial automorphism fixing either all of the spin structures or just one. The case of hyperelliptic curves and of the Klein quartic are discussed in detail.

DOI : 10.5802/afst.1251
Kallel, Sadok 1 ; Sjerve, Denis 2

1 Laboratoire Painlevé, Université de Lille I, France
2 Department of Mathematics, University of British Columbia,Canada
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Kallel, Sadok; Sjerve, Denis. Invariant Spin Structures on Riemann Surfaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 19 (2010) no. 3-4, pp. 457-477. doi : 10.5802/afst.1251. http://www.numdam.org/articles/10.5802/afst.1251/

[1] Adler (A.), Ramanan (S.).— Moduli of abelian varieties, Lecture Notes in Mathematics, 1644. Springer (1996). | MR | Zbl

[2] Aigon (A.).— Transformations Hyperboliques et Courbes Algébriques en genre 2 et 3, Thèse 2001, Université de Montpellier II. http://tel.archives-ouvertes.fr/docs/00/04/47/84/PDF/tel-00001154.pdf

[3] Arbarello (E.), Cornalba (M.), Griffiths (P.), Harris (J.).— Geometry of algebraic curves I, Spinger Grundlehren 267. | Zbl

[4] Atiyah (M.F.).— Riemann surfaces and spin structures, Ann. scient. École. Norm. Sup.(4), p. 47-62 (1971). | Numdam | MR | Zbl

[5] Bavard (C.).— La surface de Klein, le journal de maths des élèves de ENS-Lyon (http://www.ens-lyon.fr/JME/), vol.1, p. 13-22 (1993).

[6] Dolgachev (I.).— Topics in classical algebraic geometry, part I, April 20 (2006).

[7] Dolgachev (I.).— Invariant stable bundles over modular curves X(p), Recent progress in algebra (Taejon/Seoul, 1997), p. 65-99, Contemp. Math., 224, Amer. Math. Soc., Providence, RI (1999). | MR | Zbl

[8] Farkas (H.), Kra (I.).— Riemann Surfaces, Springer Graduate Texts in Math 71, 2nd Edition (1992). | MR | Zbl

[9] Johnson (D.).— Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2) 22, no. 2, p. 365-373 (1980). | MR | Zbl

[10] Masbaum (G.).— On representations of spin mapping class groups arising in spin TQFT, Geometry and physics (Aarhus, 1995), p. 197-207, Lecture Notes in Pure and Appl. Math., 184, Dekker, New York, (1997). | MR | Zbl

[11] Mukai (S.).— Plane quartics and Fano threefolds of genus twelve, The Fano Conference, p. 563-572, Univ. Torino, Turin, 2004. | MR | Zbl

[12] Mumford (D.).— Theta characteristics of an algebraic curve, Ann. Sci. École Norm. Sup. (4) 4 (1971), p. 181-192. | Numdam | MR | Zbl

[13] Nielsen J..— Die Structur periodischer Transformation von Flächen, Danske Vid. Selsk., Mat.-Fys.Medd 15 (1937), p. 1-77. | Zbl

[14] Rauch (H.E.),Lewittes (J.).— The Riemann Surface of Klein with 168 Automorphisms, Problems in analysis (papers dedicated to Solomom Bochner, 1969), 297-308. Princeton Univ. Press, Princeton, N.J., 1970 | MR | Zbl

[15] Sipe (P.L.).— Roots of the canonical bundle of the universal Teichmüller curve and certain subgroups of the mapping class group, Math. Ann. 260 (1982), no. 1, p. 67-92. | MR | Zbl

[16] Sjerve (D.).— Canonical Forms for Torsion Matrices, J. of Pure and Algebra 22 (1981) p. 103-111. | MR | Zbl

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