On Oda's problem and special loci

[BR11]

[Col12]

[CM15]

[CM23]

[DM69]

[Eke95]

[GH97]

[GM71]

[EGAI]

[EGAIII]

[Hos11]

[Hos13]

[HM11]

[Iha86]

[Iha89]

34/36

J. Bertin and M. Romagny, “Hurwitz stacks,” Mém. Soc. Math. Fr., Nouv. Sér.,

vol. 125-126, p. 219, 2011 (cit. on pp. 6, 19).

B. Collas, “Action of a Grothendieck-Teichmüller group on torsion elements of full Teichmüller modular groups of genus one,” Int. J. Number Theory, vol. 8, no. 3, pp. 763–787,

2012 (cit. on p. 3).

B. Collas and S. Maugeais, “Composantes irréductibles de lieux spéciaux d’espaces de

modules de courbes, action galoisienne en genre quelconque,” Ann. Inst. Fourier, vol. 65,

no. 1, pp. 245–276, 2015 (cit. on pp. 3–6, 21, 33).

——, “On Galois Action on the Inertia Stack of Moduli Spaces of Curves,” Publ. Res. Inst.

Math. Sci., vol. 59, no. 4, pp. 731–758, 2023 (cit. on pp. 3, 13).

P. Deligne and D. Mumford, “The irreducibility of the space of curves of given genus,”

Inst. Hautes Études Sci. Publ. Math., no. 36, pp. 75–109, 1969 (cit. on p. 7).

T. Ekedahl, “Boundary behaviour of Hurwitz schemes,” in The moduli space of curves.

Proceedings of the conference held on Texel Island, Netherlands during the last week of April

1994, Basel: Birkhäuser, 1995, pp. 173–198 (cit. on p. 6).

G. González-Diez and R. A. Hidalgo, “Conformal versus topological conjugacy of automorphisms on compact Riemann surfaces,” Bull. Lond. Math. Soc., vol. 29, no. 3, pp. 280–284,

1997 (cit. on p. 6).

A. Grothendieck and J. P. Murre, The tame fundamental groups of a formal neighbourhood

of a divisor with normal crossings on a scheme (Lect. Notes Math.). Springer, 1971, vol. 208

(cit. on pp. 24, 27, 28, 31).

A. Grothendieck, “Éléments de géométrie algébrique. I. Le langage des schémas,” Inst.

Hautes Études Sci. Publ. Math., no. 4, p. 228, 1960 (cit. on p. 23).

——, “Éléments de géométrie algébrique. III. Étude cohomologique des faisceaux cohérents,”

Inst. Hautes Études Sci. Publ. Math., no. 17, pp. 137–223, 1963 (cit. on p. 24).

Y. Hoshi, “Galois-theoretic characterization of isomorphism classes of monodromically full

hyperbolic curves of genus zero,” Nagoya Math. J., vol. 203, pp. 47–100, 2011 (cit. on p. 9).

——, “On a problem of Matsumoto and Tamagawa concerning monodromic fullness of

hyperbolic curves: Genus zero case,” Tohoku Math. J. (2), vol. 65, no. 2, pp. 231–242, 2013

(cit. on p. 3).

Y. Hoshi and S. Mochizuki, “On the combinatorial anabelian geometry of nodally nondegenerate outer representations,” Hiroshima Math. J., vol. 41, no. 3, pp. 275–342, 2011

(cit. on p. 2).

Y. Ihara, “Profinite braid groups, Galois representations and complex multiplications,” Ann.

of Math. (2), vol. 123, no. 1, pp. 43–106, 1986 (cit. on p. 2).

——, “The Galois representation arising from P1 − {0, 1, ∞} and Tate twists of even degree,”

in Galois groups over Q (Berkeley, CA, 1987), ser. Math. Sci. Res. Inst. Publ. Vol. 16,

Springer, New York, 1989, pp. 299–313 (cit. on p. 2).

Version of November 22, 2023

On Oda’s problem and special loci

[Iha94]

[IN97]

[Knu83]

[Mag+02]

[Mat96]

[MT08]

[Mor93]

[Nak96]

[Nak99]

[NTU95]

[Noo04]

[Oda93]

[RT17]

[SGA1]

[Rom11]

[Sey82]

[Tak12]

[Tak14]

[Tsu06]

̂︂ in The Grothendieck theory of dessins

——, “On the embedding of Gal(Q/Q) into GT,”

d’enfants (Luminy, 1993), ser. London Math. Soc. Lecture Note Ser. Vol. 200, With an

appendix: the action of the absolute Galois group on the moduli space of spheres with four

marked points by Michel Emsalem and Pierre Lochak, Cambridge Univ. Press, Cambridge,

1994, pp. 289–321 (cit. on p. 15).

Y. Ihara and H. Nakamura, “On deformation of maximally degenerate stable marked

curves and Oda’s problem,” J. Reine Angew. Math., vol. 487, pp. 125–151, 1997 (cit. on

pp. 2, 4, 14, 18, 19, 21, 24–26, 28, 29, 31).

F. F. Knudsen, “The projectivity of the moduli space of stable curves. II: The stacks 𝑀𝑔,𝑛 ,”

Math. Scand., vol. 52, pp. 161–199, 1983 (cit. on p. 19).

K. Magaard, T. Shaska, S. Shpectorov, and H. Völklein, “The locus of curves with

prescribed automorphism group,” in 1267, Communications in arithmetic fundamental groups

(Kyoto, 1999/2001), 2002, pp. 112–141 (cit. on p. 5).

M. Matsumoto, “Galois representations on profinite braid groups on curves,” J. Reine

Angew. Math., vol. 474, pp. 160–219, 1996 (cit. on pp. 2–4, 13, 19).

S. Mochizuki and A. Tamagawa, “The algebraic and anabelian geometry of configuration

spaces,” Hokkaido Math. J., vol. 37, no. 1, pp. 75–131, 2008 (cit. on p. 16).

S. Morita, “The extension of Johnson’s homomorphism from the Torelli group to the

mapping class group,” Invent. Math., vol. 111, no. 1, pp. 197–224, 1993 (cit. on p. 3).

H. Nakamura, “Coupling of universal monodromy representations of Galois-Teichmüller

modular groups,” Math. Ann., vol. 304, no. 1, pp. 99–119, 1996 (cit. on pp. 2, 4, 13, 34).

——, “Tangential base points and Eisenstein power series,” in Aspects of Galois theory.

Papers from the conference on Galois theory, Gainesville, FL, USA, October 14–18, 1996,

Cambridge: Cambridge University Press, 1999, pp. 202–217 (cit. on p. 14).

H. Nakamura, N. Takao, and R. Ueno, “Some stability properties of Teichmüller modular

function fields with pro-𝑙 weight structures,” Math. Ann., vol. 302, no. 2, pp. 197–213, 1995

(cit. on p. 2).

B. Noohi, “Fundamental groups of algebraic stacks,” J. Inst. Math. Jussieu, vol. 3, no. 1,

pp. 69–103, 2004 (cit. on p. 13).

T. Oda, “The universal monodromy representations on the pro-nilpotent fundamental groups

of algebraic curves,” Mathematische Arbeitstagung (Neue Serie), Max-Planck-Institut preprint

MPI/93-57, vol. 9.-15, Jun. 1993 (cit. on p. 2).

C. Rasmussen and A. Tamagawa, “Arithmetic of abelian varieties with constrained torsion,”

Trans. Amer. Math. Soc., vol. 369, no. 4, pp. 2395–2424, 2017 (cit. on p. 2).

Revêtements étales et groupe fondamental (Documents Mathématiques). Société Mathématique de France, Paris, 2003, vol. 3, pp. xviii+327, Séminaire de géométrie algébrique du Bois

Marie 1960–61, Directed by A. Grothendieck, With two papers by M. Raynaud. (cit. on

p. 7).

M. Romagny, “Connected and irreducible components in families,” Manuscr. Math., vol. 136,

no. 1-2, pp. 1–32, 2011 (cit. on p. 5).

A. Seyama, “On the curves of genus g with automorphisms of prime order 2𝑔 + 1,” Tsukuba

J. Math., vol. 6, pp. 67–77, 1982 (cit. on p. 17).

N. Takao, “Braid monodromies on proper curves and pro-ℓ Galois representations,” J. Inst.

Math. Jussieu, vol. 11, no. 1, pp. 161–181, 2012 (cit. on pp. 2, 4, 10, 13, 34).

——, “Some remarks on field towers arising from pro-nilpotent universal monodromy representations,” RIMS Kôkyûroku Bessatsu, vol. B51, pp. 55–70, 2014 (cit. on p. 2).

H. Tsunogai, “Some new-type equations in the Grothendieck-Teichmüller group arising from

geometry ℳ0,5 ,” in Primes and knots. Proceedings of an AMS special session, Baltimore, MD,

USA, January 15–16, 2003 and the 15th JAMI (Japan-US Mathematics Institute) conference,

Version of November 22, 2023

35/36

On Oda’s problem and special loci

[Uen94]

Baltimore, MD, USA, March 7–16, 2003, Providence, RI: American Mathematical Society

(AMS), 2006, pp. 263–284 (cit. on p. 15).

R. Ueno, A stability of the galois kernel arising from pro-ℓ monodromy representation, Master

thesis in RIMS, Kyoto University, 1994 (cit. on p. 34).

URL: https://www.kurims.kyoto-u.ac.jp/~bcollas/

URL: https://www.kurims.kyoto-u.ac.jp/~sphilip/

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