\[
  (1)\qquad 1 \to G_{\overline{X}} \longrightarrow \mathcal{G}_{\overline{X}}
  \longrightarrow \Gamma
\]

\[
  (2)\qquad \left\{
  \begin{array}{l}
    \Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \\
    G_{\overline{X}} = \mathrm{Aut}(\overline{X}/\overline{\mathbb{Q}})
  \end{array}\right.
\]

\[
  (5)\qquad \Sigma(\mathfrak{X}, \widetilde{\mathfrak{X}})
  \ \text{ou}\ \Sigma_{\mathfrak{X}}
  = \mathrm{Aut}(\widetilde{\mathfrak{X}} \,/\!/\, \mathfrak{X})
\]

\[
  (6)\qquad 1 \longrightarrow \pi_1(\mathfrak{X}) \longrightarrow
  \Sigma_{\mathfrak{X}} \longrightarrow \mathcal{G}_{\mathfrak{X}}
  \longrightarrow 1 ,
  \qquad \pi_1(\mathfrak{X}) = \mathrm{Aut}(\widetilde{\mathfrak{X}}/\mathfrak{X}),
  \qquad \mathcal{G}_{\mathfrak{X}} \to \Gamma
\]

\[
  (7)\qquad \xi \in \mathfrak{X}(\overline{\mathbb{Q}})
\]

\[
  \Gamma' \longrightarrow \Sigma_{\mathfrak{X}}
\]

\[
  (11)\qquad \varphi_{\xi} \in \varinjlim_{\Gamma' \subset \Gamma}
  \mathrm{Hom}(\Gamma', \Sigma_{\mathfrak{X}})
\]

\[
  (12)\qquad \text{Soit } \boxed{D = \sum_{i \in I} D_i}
\]

\[
  (13)\qquad \mathfrak{X}' = \mathfrak{X} - \operatorname{supp} D
\]

\[
  (14)\qquad 1 \to \pi_1(\mathfrak{X}') \longrightarrow
  \Sigma_{\mathfrak{X}'} \longrightarrow \mathcal{G}_{\mathfrak{X}'}
  \to 1 .
\]

\[
  (16)\qquad \mathcal{G}_{\mathfrak{X}'} \longrightarrow \mathfrak{S}_I
\]

\[
  (17)\qquad T(\overline{\mathbb{Q}}) \xrightarrow{\ \varphi_i\ }
  \pi_1(\mathfrak{X}'),
  \qquad T(\overline{\mathbb{Q}}) = \varprojlim \mu_n(\overline{\mathbb{Q}}),
  \qquad \varphi_i \in \mathrm{Homext}(T(\overline{\mathbb{Q}}), \pi_1(\mathfrak{X}'))
\]

\[
  (18)\qquad (\mathcal{G}_{\mathfrak{X}'} \to)\ \Gamma
  \xrightarrow{\ \chi\ } \widehat{\mathbb{Z}}^{*} \xrightarrow{\ \sim\ }
  \mathrm{Aut}(T(\overline{\mathbb{Q}}))
\]

\[
  (19)\qquad \mathcal{G}_{\mathfrak{X}'} \longrightarrow
  \mathrm{Autext}\bigl(\pi_1(\mathfrak{X}'), T(\overline{\mathbb{Q}}), (\varphi_i)\bigr).
\]

\[
  \widehat{\widetilde{\mathfrak{X}}}{}' \longrightarrow \mathfrak{X}
\]

\[
  (20)\qquad \widetilde{\mathfrak{X}}'(I) \longrightarrow I
\]

\[
  (21)\qquad \widetilde{\mathfrak{X}}'(i) \subset \widetilde{\mathfrak{X}}'(I)
\]

\[
  (22)\qquad \widetilde{\mathfrak{X}}'(i) \xrightarrow{\ \Phi_i\ }
  \mathrm{Hom}(T(\overline{\mathbb{Q}}), \pi_1(\mathfrak{X}'))
\]

\[
  (23)\qquad \widetilde{\mathfrak{X}}'(I) \xrightarrow{\ \Phi\ }
  \mathrm{Hom}(T(\overline{\mathbb{Q}}), \pi_1(\mathfrak{X}'))
\]

\[
  (24)\qquad \widetilde{\mathfrak{X}}'(J) \longrightarrow J
\]

\[
  (25)\qquad \Psi = (\Psi_j)_{j \in J} : \bigl(\widetilde{\mathfrak{X}}(j)
  \longrightarrow \mathrm{Hom}(\mathcal{G}_j, \Sigma')\bigr)
\]

\[
  \mathrm{I}\quad \left\{
  \begin{array}{l}
    \mathfrak{X} \text{ de t.f.\ sur } \overline{\mathbb{Q}} \\
    D = \sum_{i \in I} D_i \text{ diviseur effectif sur } \mathfrak{X}
      \text{ à comp.\ simples, } \mathfrak{X} \text{ lisse en les pts} \\
    \qquad \text{maximaux de } \operatorname{supp} D \\
    J \subset \mathfrak{X}'(\overline{\mathbb{Q}}), \text{ où }
      \mathfrak{X}' = \mathfrak{X} - D
  \end{array}\right.
\]

\[
  \mathrm{II}\quad \left\{
  \begin{array}{l}
    \mathcal{G}' = \text{groupe des autom.\ du schéma absolu } \mathfrak{X}
      \text{ qui invarie } D \text{ et } J \\
    G' = \text{s-gp de } \mathcal{G}' \text{ formé des }
      \overline{\mathbb{Q}}\text{-autom.} \\
    T(\overline{\mathbb{Q}}) = \varprojlim_n \mu_n(\overline{\mathbb{Q}}) \\
    \Gamma = \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})
  \end{array}\right.
\]

\[
  (1)\qquad 1 \longrightarrow G' \longrightarrow \mathcal{G}' \longrightarrow \Gamma
\]

\[
  \pi_1(\mathfrak{X}') = \mathrm{Aut}(\widetilde{\mathfrak{X}}'/\mathfrak{X}')
\]

\[
  (3)\qquad \text{str.\ d'extension}\quad
  1 \to \pi_1(\mathfrak{X}') \longrightarrow \Sigma' \longrightarrow
  \mathcal{G}' \to 1
\]

\[
  (4)\qquad \Sigma'\text{-ensembles}\quad \left\{
  \begin{array}{l}
    \widetilde{\mathfrak{X}}'(I) \\
    \widetilde{\mathfrak{X}}'(J)
  \end{array}\right.
\]

\[
  (5)\qquad \text{Opérations de } \Sigma' \text{ sur } I, J \text{ via }
  \mathcal{G}', \text{ et } \Sigma'\text{-applications}\quad \left\{
  \begin{array}{l}
    \widetilde{\mathfrak{X}}'(I) \to I \\
    \widetilde{\mathfrak{X}}'(J) \to J \simeq \widetilde{\mathfrak{X}}'(J)/\pi_1
  \end{array}\right.
\]

\[
  (6)\qquad \text{application}\quad \Phi :
  \widetilde{\mathfrak{X}}'(I) \longrightarrow
  \mathrm{Hom}(T(\overline{\mathbb{Q}}), \pi_1(\mathfrak{X}'))
\]

\[
  (7)\qquad \text{Syst.\ d'application}\quad
  \Psi = (\Psi_j)_{j \in J} : \bigl(\widetilde{\mathfrak{X}}'(j)
  \longrightarrow \mathrm{Splitt}(\Sigma'_j / \mathcal{G}'_j)\bigr)_{j \in J}
\]

\[
  ({}^{g}u)(\sigma) = g\, u(g_1^{-1} \sigma g_1)\, g^{-1}
\]

\[
  \Gamma' \longrightarrow \mathcal{G}'
\]

\[
  (9)\qquad 1 \to \pi_1(\mathfrak{X}') \longrightarrow \Sigma'_{\Gamma'}
  \longrightarrow \Gamma' \to 1
\]

\[
  (10)\qquad E \longmapsto \mathcal{B}_{\gamma} \wedge^{\pi_1} E
\]

\[
  (11)\qquad \left\{
  \begin{array}{l}
    \mathcal{Y}'(I) \simeq ({}^{\gamma}\mathcal{Y}')(I) \\
    \mathcal{Y}'(J) = ({}^{\gamma}\mathcal{Y}')(J).
  \end{array}\right.
\]

\[
  \begin{array}{l}
    \mathcal{Y}'(I) \simeq \widetilde{\mathfrak{X}}'(I) \wedge^{\pi_1} E \\
    \mathcal{Y}'(J) \simeq \widetilde{\mathfrak{X}}'(J) \wedge^{\pi_1} E
  \end{array}
\]

\[
  \left\{
  \begin{array}{l}
    \widetilde{\mathfrak{X}}'(I) \wedge^{\pi_1} E \simeq
      (\widetilde{\mathfrak{X}}'(I) \wedge^{\pi_1} \mathcal{B}_{\gamma}) \wedge^{\pi_1} E \\
    \widetilde{\mathfrak{X}}'(J) \wedge^{\pi_1} E \simeq
      (\widetilde{\mathfrak{X}}'(J) \wedge^{\pi_1} (\mathcal{B}_{\gamma})) \wedge^{\pi_1} E
  \end{array}\right.
\]

\[
  \left\{
  \begin{array}{l}
    \widetilde{\mathfrak{X}}'(I) \simeq \widetilde{\mathfrak{X}}'(I) \wedge^{\pi_1} \mathcal{B}_{\gamma} \\
    \widetilde{\mathfrak{X}}'(J) \simeq \widetilde{\mathfrak{X}}'(J) \wedge^{\pi_1} \mathcal{B}_{\gamma}
  \end{array}\right.
\]

\[
  \left\{
  \begin{array}{l}
    \mathcal{B}_{\gamma} \longrightarrow \mathrm{Aut}(\widetilde{\mathfrak{X}}'(I))
      = \mathrm{Bij}(\widetilde{\mathfrak{X}}'(I), \widetilde{\mathfrak{X}}'(I)) \\
    \quad \text{et itou pour } \widetilde{\mathfrak{X}}'(J))
  \end{array}\right.
\]

\[
  \Sigma'_{\Gamma'} = \bigcup_{\gamma \in \Gamma'} \mathcal{B}_{\gamma}
  \longrightarrow \mathrm{Aut}_{\mathrm{ens}}(\widetilde{\mathfrak{X}}'(I)),
  \qquad \longrightarrow \mathrm{Aut}_{\mathrm{ens}}(\widetilde{\mathfrak{X}}'(J))
\]

\[
  \begin{array}{l}
    \Psi^{\mathrm{ext}} : \widetilde{\mathfrak{X}}'(J) \longrightarrow
      \mathrm{Splitt}(\Sigma'/\mathcal{G}')/\pi_1 \\
    \Psi_j : (\widetilde{\mathfrak{X}}'(j) \longrightarrow
      \mathrm{Splitt}(\Sigma'_j/\mathcal{G}'_j))
  \end{array}
\]

\[
  \widetilde{\mathfrak{X}}' \xrightarrow{\ \psi_{\widetilde{\xi}}\ }
  \widetilde{\mathfrak{X}}'[j]
\]

\[
  \psi_{\gamma \cdot \widetilde{\xi}} = \psi_{\widetilde{\xi}} \circ
  \gamma_{\widetilde{\mathfrak{X}}'}
\]

\[
  \widetilde{\mathfrak{X}}'[j] \xleftarrow{\ \sim\ }
  \underbrace{\widetilde{\mathfrak{X}}'(j)}_{\text{torseur à dr.\ sous } \pi_1}
  \wedge^{\pi_1} \widetilde{\mathfrak{X}}'
\]

\[
  \begin{aligned}
    \mathrm{Isom}_{\Pi_1}(j, j') &\overset{\mathrm{df}}{=}
      \mathrm{Isom}(\widetilde{\mathfrak{X}}'[j], \widetilde{\mathfrak{X}}'[j']) \\
    &\simeq \mathrm{Isom}_{\pi_1\text{-tors.\ à droite}}
      (\widetilde{\mathfrak{X}}'(j), \widetilde{\mathfrak{X}}'(j'))
  \end{aligned}
\]

\[
  \begin{array}{ccc}
    \mathrm{Isom}_{\Pi_1}(j, j') & \xrightarrow{\ g_{\Pi_1}\ } &
      \mathrm{Isom}(gj, gj') \\
    \| & & \| \\
    \mathrm{Isom}_{\pi_1}(\widetilde{\mathfrak{X}}'(j), \widetilde{\mathfrak{X}}'(j'))
      & \overset{?}{\simeq} &
      \mathrm{Isom}_{\pi_1}(\widetilde{\mathfrak{X}}'(gj), \widetilde{\mathfrak{X}}'(gj'))
  \end{array}
\]

\[
  \mathcal{B}_g \subset \Sigma'\quad \text{image inverse de } g \text{ par }
  \Sigma' \to \mathcal{G}'
\]

\[
  \begin{array}{l}
    \widetilde{\mathfrak{X}}'(gj) \simeq \widetilde{\mathfrak{X}}'(j) \wedge^{\pi_1} \mathcal{B}_g \\
    \widetilde{\mathfrak{X}}'(gj') \simeq \widetilde{\mathfrak{X}}'(j') \wedge^{\pi_1} \mathcal{B}_g
  \end{array}
\]

\[
  \left\{
  \begin{array}{ll}
    1 \to \pi \longrightarrow \Sigma \longrightarrow G \to 1 &
      \text{ext.\ de groupes} \\
    \mathfrak{Z}\ \ \Sigma\text{-ens.}, \neq \emptyset,\ \text{où } \pi \subset
      \Sigma \text{ opère librement}
  \end{array}\right.
\]

\[
  C \mathrel{\substack{\xrightarrow{\ \alpha\ } \\ \xleftarrow[\ \beta\ ]{}}} C'
\]

\[
  \alpha(\Sigma, G, \pi, \mathfrak{Z}\ldots) = (\Pi, \widetilde{\Pi}, p, G, \sim)
\]

\[
  \left\{
  \begin{array}{l}
    \mathrm{Ob}\,\Pi = \mathfrak{Z}/\pi \quad (\text{soit } J) \\
    \mathrm{Isom}(j, j') = \mathrm{Isom}_{\pi}(\mathfrak{Z}_j, \mathfrak{Z}_{j'}) \\
    \text{composition des flèches évidente}
  \end{array}\right.
  \qquad \text{d'où groupoïde } \Pi
\]

\[
  \begin{array}{l}
    \mathrm{Isom}(z, z') = \{\mathrm{pt}\} \quad \forall z, z' \in \mathfrak{Z}
      \quad (\widetilde{\Pi} \text{ cat.\ discrète\ldots}) \\
    \qquad\downarrow \\
    \mathrm{Isom}(j, j') \simeq \mathrm{Isom}(\mathfrak{Z}_j, \mathfrak{Z}_{j'})
      \quad (\text{si } j = \bar z,\ j' = \bar z')
  \end{array}
\]

\[
  \beta(\Pi, \widetilde{\Pi}, p, G, \sim) = (\Sigma, G, \pi, \mathfrak{Z})
  \quad \text{ainsi :}
\]

\[
  \pi = \mathrm{Aut}(\widetilde{\Pi}/\Pi)
\]

\[
  1 \to \pi \longrightarrow \Sigma \longrightarrow G \to 1
\]

\[
  \Phi : \mathcal{T} \longrightarrow \mathrm{Hom}_{\mathrm{gr}}(T, \pi)
\]

\[
  J = \mathcal{T}/\pi
\]

\[
  \begin{array}{l}
    \mathrm{Ob}\,\langle T, J \rangle = J \\
    \mathrm{Isom}(j, j') = \left\{
      \begin{array}{ll} \emptyset & \text{si } j \neq j' \\
        T & \text{si } j = j' \end{array}\right.
  \end{array}
\]

\[
  \langle T, J \rangle \longrightarrow \Pi^{\wedge}
\]

\[
  \Phi(t) : T \longrightarrow \pi
\]

\[
  \pi_0 \langle \mathcal{T}, \pi \rangle = \mathcal{T}/\pi \overset{\mathrm{df}}{=} J
\]

\[
  \pi_1(\mathcal{T}, t) = \pi_t \quad (\text{stabilisateur de } t \text{ dans } \pi)
\]

\[
  \Phi_t : T \xrightarrow{\ \sim\ } \pi_t
\]

\[
  \pi_{t,t'} \;\simeq\; \mathrm{Isom}_{\langle \mathcal{T}, \pi \rangle}(t, t')
\]

\[
  \langle \mathcal{T}, \pi \rangle \longrightarrow
  \text{Cat des revêt.\ universels de } \Pi
\]

\[
  \Lambda \xrightarrow{\ \text{foncteur fidèle}\ } \text{groupoïde}
\]

\[
  \underbrace{\mathrm{Isom}_{\Lambda}(t, t')}_{\pi_{t,t'}} \hookrightarrow \pi
\]

\[
  \left\{
  \begin{array}{l}
    \forall t \in \mathcal{T} = \mathrm{Ob}\,\Lambda, \text{ la famille des }
      \pi_{t,t'} \subset \pi \ (t' \in \mathcal{T} \text{ variable}) \text{ forme une}\\
    \text{pseudo-partition de } \pi, \text{ i.e. } \forall \lambda \in \pi,\
      \exists \text{ un unique } t' \text{ tel que } \lambda \in \pi_{t,t'},\\
    \text{i.e. une unique flèche de } \Lambda \text{ au-dessus de } \lambda
      \text{ qui soit d'origine } t.
  \end{array}
  \right.
\]

\[
  \Lambda \simeq \langle \mathcal{T}, \pi \rangle
\]

\[
  F({}^{\gamma}\Pi') \simeq F(\Pi') \qquad (\Pi' \in \ill{}\ \Pi)
\]

\[
  B_{\mathrm{PGL}(2)}, \quad B_{\mathrm{Aff}(1)}, \quad B_{\mathbb{G}_m},
\]

\[
  \pi_1(\mathcal{T}_{0,\nu}) = 0, \qquad 0 \leqslant \nu \leqslant 3 .
\]

\[
  \mathbb{P}^1 \smallsetminus \{0, 1, \infty\} = \mathbb{E}^1 \smallsetminus \{0, 1\} .
\]

\[
  (*) \qquad \pi_1(M_{g,\nu;\xi}) \longrightarrow
  \mathrm{Autext}_{\mathrm{lac}}^{\circ}\bigl(\pi_1(\overline{X} \smallsetminus
  \{x_1, \ldots, x_\nu\})\bigr)
\]

\[
  (**) \qquad \pi_1(\overline{M}_{g,\nu;\xi}) \longrightarrow
  \mathrm{Autext}_{\mathrm{lac}}^{\circ\circ}\bigl(\pi_1(\overline{X} \smallsetminus
  \{x_1, \ldots, x_\nu\})\bigr)
\]

\[
  (***) \qquad \pi_1(M_{g,\nu;\xi}) \longrightarrow
  \mathrm{Autext}_{\mathrm{lac}}^{\circ\circ}\,
  \underbrace{\pi_1(X \smallsetminus \{x_1, \ldots, x_\nu\})}_{=\ \mathcal{T}_{g,\nu}\ (\text{Teichmüller})}
\]

\[
  \underbrace{\mathrm{Autext}_{\mathrm{lac}}^{\circ}\bigl(\pi_1(X \smallsetminus
  \{x_1, \ldots, x_\nu\})^{\wedge}\bigr)}_{\text{groupe \ill{} bien connu}}
  \big/ \widehat{\mathcal{T}}_{g,\nu}
\]

\[
  \mathrm{Autext}_{\mathrm{lac}}^{\circ}\bigl(\pi_1(\mathbb{P}^1 \smallsetminus
  \{0, 1, \infty\})\bigr).
\]

\[
  (5) \qquad 1 \to \pi_1(\overline{M}_{g,\nu}) \longrightarrow
  \pi_1(N_{g,\nu;\xi}) \longrightarrow \mathfrak{S}_\nu \times \Gamma \to 1
\]

\[
  \pi_1(N_{g,\nu;\xi}) = \pi_1(\overline{N}_{g,\nu}, \Gamma; \xi) =
  \pi_1(M_{g,\nu}; \mathfrak{S}_\nu, \xi) = \pi_1(\overline{M}_{g,\nu},
  \mathfrak{S}_\nu \times \Gamma)
\]

\[
  \pi_1(N_{g,\nu;\xi}) \xrightarrow{\ \sim\ }
  \mathrm{Autext}_{\mathrm{lac}}\bigl(\pi_1(X_\xi \smallsetminus \{x_1, \ldots, x_\nu\})\bigr)
\]

\[
  \Gamma \simeq \mathrm{Autext}_{\mathrm{lac}}\bigl(\pi_1(X_\xi \smallsetminus
  \{x_1, \ldots, x_\nu\})\bigr) \big/ \widehat{\widetilde{\mathcal{T}}}_{g,\nu}
\]

\[
  \widetilde{\mathcal{T}}_{g,\nu} \simeq
  \mathrm{Autext}_{\mathrm{lac}}^{+}\bigl(\pi_1(X_\xi^{\mathrm{top}} \smallsetminus
  \{x_1, \ldots, x_\nu\})\bigr).
\]

\[
  \Gamma \longrightarrow
  \mathrm{Autext}\bigl(\underbrace{\widehat{\mathcal{T}}_{g,\nu}}_{\pi_1(\overline{M}_{g,\nu})}\bigr)
  \quad
  \left\{\begin{array}{l} \text{injectif} \\ \text{bijectif} \end{array}\right\} ?
\]

\[
  M'_{g,\nu}(k) \ni \xi, \qquad \mathcal{T}'_{g,\nu} \ni G_\xi
\]

\[
  \widehat{\widetilde{\Sigma}}_{g,\nu} = \overline{\Sigma}_{g,\nu}
  \qquad\qquad \Sigma_{g,\nu}
\]