Composantes irréductibles de lieux spéciaux d’espaces de modules de courbes, action galoisienne en genre quelconque

Collas, Benjamin; Maugeais, Sylvain

Annales de l'Institut Fourier, Tome 65 (2015) no. 1, pp. 245-276.

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 $2$.

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.

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

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

[3] Catanese, F. Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of $Sing\left({\overline{𝔐}}_g\right)$, Advances in geometric analysis, Volume 21, Int. Press, Somerville, MA, 2012, p. 281-306, arXiv :1011.0316v1 | MR

[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, Volume 84 (2012) no. 3, pp. 763-787 | DOI | MR | Zbl

[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, Volume 24 (2012) no. 3, pp. 605-622 | DOI | EuDML | Numdam | MR | Zbl

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

[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), Volume 187 (2008) no. 1, pp. 185-186 | DOI | MR | Zbl

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

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

[10] Drinfelʼd, V. G. On quasitriangular quasi-Hopf algebras and on a group that is closely connected with $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$, Algebra i Analiz, Volume 2 (1990) no. 4, pp. 149-181 | MR | Zbl

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

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

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

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

[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., Volume 173 (1992), pp. 75-93 | Zbl

[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

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

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

[19] Ihara, Y.; Schneps, L.; Lochak, P. On the embedding of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ into $\widehat{GT}$, The Grothendieck Theory of Dessins d’Enfants, Volume 200, Cambridge Univ. Press, Cambridge, 1994, pp. 289-321 | MR

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

[21] Knudsen, F. F. The projectivity of the moduli space of stable curves. II. The stacks ${\mathrm{M}}_{g,n}$, Math. Scand., Volume 52 (1983) no. 2, pp. 161-199 | MR | Zbl

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

[23] Lochak, P.; Schneps, L.; symp, proc. Open problems in Grothendieck-Teichmüller theory, Amer. Math. Soc., 2006, pp. 165-186 | MR | Zbl

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

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

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

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

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

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

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

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

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

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

[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 | Zbl

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

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

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