Composantes irréductibles de lieux spéciaux d’espaces de modules de courbes, action galoisienne en genre quelconque  [ Irreducible components of special loci in moduli spaces of curves, Galois action in general genus ]
Annales de l'Institut Fourier, Volume 65 (2015) no. 1, p. 245-276

In this paper we characterise the action of the absolute Galois group on the geometric finite cyclic groups without étale factorization of stack inertia of the profinite geometric fundamental group of moduli spaces of marked curves. As a complementary result, we give the same action on prime order profinite elements in genus 2.

Dans cet article, nous caractérisons l’action du groupe de Galois absolu sur les groupes d’inertie champêtre géométriques cycliques et sans factorisation étale du groupe fondamental géométrique des espaces de modules de courbes marquées. Nous établissons par ailleurs la même action sur les éléments de torsion profinis d’ordre premier en genre 2.

DOI : https://doi.org/10.5802/aif.2930
Classification:  11R32,  14H10,  14H30,  14H45
Keywords: algebraic fundamental group, stack inertia, special loci, good groups
@article{AIF_2015__65_1_245_0,
     author = {Collas, Benjamin and Maugeais, Sylvain},
     title = {Composantes irr\'eductibles de lieux sp\'eciaux d'espaces de modules de courbes, action galoisienne en genre quelconque},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {65},
     number = {1},
     year = {2015},
     pages = {245-276},
     doi = {10.5802/aif.2930},
     zbl = {1326.11069},
     language = {fr},
     url = {http://www.numdam.org/item/AIF_2015__65_1_245_0}
}
Collas, Benjamin; Maugeais, Sylvain. Composantes irréductibles de lieux spéciaux d’espaces de modules de courbes, action galoisienne en genre quelconque. Annales de l'Institut Fourier, Volume 65 (2015) no. 1, pp. 245-276. doi : 10.5802/aif.2930. http://www.numdam.org/item/AIF_2015__65_1_245_0/

[1] Bertin, J.; Romagny, M. Champs de Hurwitz, SMF, Mémoire de la SMF, Tome 125-126 (2011) (arXiv :math/0701680v1) | Numdam | MR 2920693 | Zbl 1242.14025

[2] Broughton, S. A. The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology Appl., Tome 37 (1990) no. 2, pp. 101-113 | Article | MR 1080344 | Zbl 0747.32017

[3] Catanese, F. Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of Sing(𝔐 ¯ g ), Advances in geometric analysis, Int. Press, Somerville, MA, Tome 21 (2012), p. 281-306, arXiv :1011.0316v1 | MR 3077261

[4] Collas, B. Action of a Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups of genus one, International Journal of Number Theory, Tome 84 (2012) no. 3, pp. 763-787 | Article | MR 2904929 | Zbl 1288.14015

[5] Collas, B. Action of the Grothendieck-Teichmüller group on torsion elements of mapping class groups in genus zero, Journal de Théorie des Nombres de Bordeaux, Tome 24 (2012) no. 3, pp. 605-622 | Article | Numdam | MR 3010631 | Zbl 1278.14040

[6] Cornalba, M. On the locus of curves with automorphisms, Ann. Mat. Pura Appl. (4), Tome 149 (1987), pp. 135-151 | Article | MR 932781 | Zbl 0649.14013

[7] Cornalba, M. Erratum : “On the locus of curves with automorphisms” [Ann. Mat. Pura Appl. (4) 149 (1987), 135–151], Ann. Mat. Pura Appl. (4), Tome 187 (2008) no. 1, p. 185-186 | Article | MR 932781 | Zbl 1150.14003

[8] Cui, Y. Special loci in moduli of marked curves, Michigan Math. J., Tome 56 (2008), pp. 495-512 | Article | MR 2488722 | Zbl 1162.14016

[9] Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Publications Mathématiques de l’IHES, Tome 36 (1969) no. 1, pp. 75-109 | Article | Numdam | MR 262240 | Zbl 0181.48803

[10] DrinfelʼD, V. G. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with Gal( ¯/), Algebra i Analiz, Tome 2 (1990) no. 4, pp. 149-181 | MR 1080203 | Zbl 0718.16034

[11] Dèbes, P.; Douai, J.-C. Algebraic covers : field of moduli versus field of definition, Annales Sci. E.N.S, Tome 30 (1997), pp. 303-338 | Numdam | MR 1443489 | Zbl 0906.12001

[12] Ekedahl, T. Boundary behaviour of Hurwitz schemes, The moduli space of curves (Texel Island, 1994), Birkhäuser Boston, Boston, MA (Progr. Math.) Tome 129 (1995), pp. 173-198 | MR 1363057 | Zbl 0862.14018

[13] Frediani, P.; Neumann, F. Étale Homotopy Types of Moduli Stacks of Algebraic Curves with Symmetries, K-Theory (2003) no. 30, pp. 315-340 | Article | MR 2064243 | Zbl 1059.14027

[14] Fried, M. Fields of definition of function fields and Hurwitz families—groups as Galois groups, Comm. Algebra, Tome 5 (1977) no. 1, pp. 17-82 | Article | MR 453746 | Zbl 0478.12006

[15] Gonzalez-Diez, G.; Harvey, W. Fields of definition of function fields and Hurwitz families—groups as Galois groups, London Math. Soc. Lect. Note Ser., Tome 173 (1992), pp. 75-93 | Zbl 0763.32014

[16] Grothendieck, A. Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas IV, Inst. Hautes Études Sci. Publ. Math. (1967) no. 32 | Numdam | Zbl 0135.39701

[17] Grothendieck, A.; Lochak, P.; Schneps, L. Esquisse d’un Programme, Geometric Galois Actions I, Cambridge Univ. Press, Cambridge, Tome 242 (1997), pp. 5-48 | MR 1483107 | Zbl 0901.14001

[18] Grothendieck, A.; Murre, J. P. The Tame Fundamental Group of a Formal Neighbourhood of a Divisor with Normal Crossings on a Scheme, Springer-Verlag, New York, Lecture Notes in Mathematics, Tome 208 (1971) | MR 316453 | Zbl 0216.33001

[19] Ihara, Y.; Schneps, L.; Lochak, P. On the embedding of Gal( ¯/) into GT ^, The Grothendieck Theory of Dessins d’Enfants, Cambridge Univ. Press, Cambridge, Tome 200 (1994), pp. 289-321 | MR 1305402

[20] Kerckhoff, S. P. The Nielsen realization problem, Ann. of Math. (2), Tome 117 (1983) no. 2, pp. 235-265 | Article | MR 690845 | Zbl 0528.57008

[21] Knudsen, F. F. The projectivity of the moduli space of stable curves. II. The stacks M g,n , Math. Scand., Tome 52 (1983) no. 2, pp. 161-199 | MR 702953 | Zbl 0544.14020

[22] Lochak, P. Results and conjectures in profinite Teichmüller theory, Galois-Teichmüller theory and arithmetic geometry, Math. Soc. Japan, Tokyo (Adv. Stud. Pure Math.) Tome 63 (2012), pp. 263-335 | MR 3051247

[23] Lochak, P.; Schneps, L.; Symp, Proc. Open problems in Grothendieck-Teichmüller theory, Amer. Math. Soc. (2006), pp. 165-186 | MR 2264540 | Zbl 1222.14046

[24] Maugeais, S. Quelques déformations sur les déformations équivariantes des courbes stables, Manuscripta Math. (2006) no. 120, pp. 53-82 | Article | MR 2223481 | Zbl 1101.14038

[25] Mumford, D. Abelian quotients of the Teichmüller modular group, Journal d’Analyse Mathématique, Tome 18 (1967) no. 1, pp. 227-244 | Article | MR 219543 | Zbl 0173.22903

[26] Nakamura, H. Galois rigidity of pure sphere braid groups and profinite calculus, Journal Mathematical Sciences University Tokyo, Tome 1 (1994), pp. 71-136 | MR 1298541 | Zbl 0901.14012

[27] Nakamura, H. Galois representations in the profinite Teichmüller modular groups, Geometric Galois actions, 1, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Series) (1997), pp. 159-174 | MR 1483116 | Zbl 0911.14014

[28] Nakamura, H. Limits of Galois representations in fundamental groups along maximal degeneration of marked curves. I, Amer. J. Math., Tome 121 (1999) no. 2, pp. 315-358 | Article | MR 1680325 | Zbl 1006.12001

[29] Nakamura, H.; Schneps, L. On a subgroup of the Grothendieck-Teichmüller group acting on the tower of profinite Teichmüller modular groups, Inventiones mathematica, Tome 141 (2000) no. 1, pp. 503-560 | Article | MR 1779619 | Zbl 1077.14030

[30] Noohi, B. Fundamental groups of algebraic stacks, Journal of the Institute of Mathematics of Jussieu, Tome 3 (2004) no. 01, pp. 69-103 | Article | MR 2036598 | Zbl 1052.14001

[31] Oda, T. Etale homotopy type of the moduli spaces of algebraic curves, Geometric Galois actions, 1, Cambridge Univ. Press, Cambridge (London Math. Soc. Lecture Note Ser.) Tome 242 (1997), pp. 85-95 | MR 1483111 | Zbl 0902.14019

[32] Romagny, M. Composantes connexes et irréductibles en familles, Manuscripta Math., Tome 136 (2011) no. 1-2, pp. 1-32 | Article | MR 2820394 | Zbl 1266.14010

[33] Schneps, L. Special loci in moduli spaces of curves, Galois groups and fundamental groups, Cambridge Univ. Press, Cambridge, Tome 41 (2003) | MR 2012218 | Zbl 1071.14028

[34] Serre, J.-P. Two letters on non-abelian cohomology, Geometric galois actions : around Grothendieck’s esquisse d’un programme, Cambridge Univ. Press (1997) | MR 1483117 | Zbl 0886.20035

[35] Symonds, P. On cohomology isomorphisms of groups, J. Algebra, Tome 313 (2007) no. 2, pp. 802-810 | Article | MR 2329570 | Zbl 1131.20038

[36] Tufféry, S. Déformations de courbes avec action de groupe, Forum Math., Tome 5 (1993) no. 3, pp. 243-259 | MR 1216034 | Zbl 0809.14005

[37] Zoonekynd, V. La tour de Teichmüller-Grothendieck, Institut de Mathématiques de Jussieu (2001) (Ph. D. Thesis)