\begin{equation*}
(1)\quad
\begin{cases}
I = \widehat{X} \setminus X \subset \widehat{X}^{\mathrm{lisse}}(k) & \text{« pts à l'infini » de } X\\
A = X \setminus X^{\mathrm{lisse}} = \widehat{X} \setminus \widehat{X}^{\mathrm{lisse}} \subset X(k) & \text{(pts singuliers de } X\text{)}
\end{cases}
\end{equation*}

\[
\widetilde{I} = \widehat{\widetilde{X}} \setminus \widetilde{X} = \text{Image inverse de } I \text{ par } \widehat{\widetilde{X}} \to \widehat{X}
\]

\[
(2)\qquad \widetilde{I} \xrightarrow{\ \sim\ } I
\]

\[
(3)\qquad \widetilde{A} \longrightarrow A
\]

\[
(6)\qquad (Y, \widetilde{A}, I, \sigma_{\widetilde{A}}) \longmapsto X
\]

\[
(7)\qquad S = \pi_{0}(Y) \simeq \text{ens. des comp. irréd. de } X
\]

\[
(9)\qquad I \longrightarrow S
\]

\[
I \longrightarrow S
\]

\[
S \xrightarrow{\ g\ } \mathbb{N}, \qquad \alpha \longmapsto \text{genre de } Y_{\alpha} = \widehat{\widetilde{X}}_{\alpha}.
\]

\[
(11)\qquad \pi_{0}(G_{X}) \simeq \pi_{0}(X),
\]

\begin{equation*}
(12)\quad
\begin{cases}
g(\widehat{X}) = \dim H^{1}(\widehat{X}, \mathcal{O}_{\widehat{X}})\\
\chi_{\mathrm{coh}}(\widehat{X}) = \chi(X, \mathcal{O}_{X}) = 1 - g(X)
\end{cases}
\end{equation*}

\[
(13)\qquad \chi_{\mathrm{coh}}(\widehat{X}) = \chi(\widehat{\widetilde{X}}) - \underbrace{\operatorname{card} A}_{\mu} = \sum_{\alpha \in S} (1 - g_{\alpha}) - \mu
\]

\begin{equation*}
(14)\quad
\begin{aligned}
g(\widehat{X}) &= \sum g_{\alpha} + (-\operatorname{card} S + \underbrace{\operatorname{card} A}_{\mu} + 1)\\
&= \sum g_{\alpha} + h_{1}
\end{aligned}
\end{equation*}

\[
(15)\qquad h_{1} = \operatorname{rg}_{\mathbb{Z}} H_{1}(G_{X}, \mathbb{Z}) .
\]

\[
\chi(G_{X}) = \underbrace{h_{0}}_{1} - h_{1} = \operatorname{card} S - \operatorname{card} A
\]

\[
1 - \operatorname{card} S + \operatorname{card} A = h_{1}, \qquad \text{O.K.}
\]

\[
g(\widehat{X}) = \underbrace{\sum_{\alpha \in S} g(\alpha) + h_{1}}_{\overset{\mathrm{d\acute ef}}{=}\ \text{genre de } G}
\]

\[
(16)\qquad \nu_{\alpha} = \operatorname{card} I_{\alpha} ,
\]

\[
(17)\qquad \widehat{\nu}_{\alpha} = \nu_{\alpha} + \omega_{\alpha}
\]

\[
g = 1 = \sum g_{\alpha} + h_{1}
\]

\[
g = 0 = \sum g_{\alpha} + h_{1}
\]

\[
\nu \geq 2 + 2 = 4, \quad \text{absurde.}
\]

\[
(\underline{S}, \underline{\widetilde{A}}, \sigma_{\underline{\widetilde{A}}}, \underline{I}, \underline{\widetilde{A}} \to \underline{S}, \underline{I} \to \underline{S}, \underline{S} \xrightarrow{g} \mathbb{N}_{S})
\]

\[
(18)\qquad \Gamma = \operatorname{Aut} G
\]

\[
(20)\qquad \mathcal{M}_{G}(\mathcal{S})
\]

\[
(21)\qquad \mathcal{M}_{G}(\mathcal{S}) \approx \prod_{\alpha \in S} \mathcal{M}_{g_{\alpha}, \widehat{I}_{\alpha}}(\mathcal{S}) .
\]

\[
(22)\qquad \mathcal{S} \longmapsto \mathcal{M}_{G}(\mathcal{S})
\]

\[
\mathcal{S} \longmapsto \mathcal{M}_{g_{\alpha}, \widehat{I}_{\alpha}}(\mathcal{S}) ,
\]

\[
(23)\qquad \mathcal{M}_{G} \simeq \prod_{\alpha} \mathcal{M}_{g_{\alpha}, \widehat{I}_{\alpha}} .
\]

\[
(24)\qquad (\mathcal{M}_{G}, \Gamma) = \mathcal{M}_{[G]} ,
\]

\[
\text{(24)}\qquad \mathcal{S} \longmapsto \hat{M}_{g,\nu}(\mathcal{S}) = \text{MD-courbes relatives sur } \mathcal{S}\text{, de type numérique égal à } (g,\nu)
\]

\[
\text{(25)}\qquad M_{[G]} \hookrightarrow \hat{M}_{g,\nu}
\]

\[
\text{(26)}\qquad M_{g,\nu} \hookrightarrow \hat{M}_{g,\nu} .
\]

\[
M_G \simeq \prod_\alpha \underbrace{M_{g_\alpha, \hat{I}_\alpha}}_{\dim\, 3(g_\alpha - 1) + \hat{\nu}_\alpha}
\]

\begin{align*}
\text{(27)}\quad \dim M_G/\mathbb{Z}
  &= \sum_{\alpha\in S} 3(g_\alpha - 1) + \sum \hat{\nu}_\alpha \\
  &= 3\sum g_\alpha - 3\underbrace{\operatorname{card} S}_{s_0} + \sum_{\alpha\in S_1} \nu_\alpha + \sum_{\alpha\in S_1} \tilde{\mu}_\alpha \\
  &= 3\sum g_\alpha - 3s_0 + \nu + 2s_1
\end{align*}

\[
-3s_0 + 2s_1 = -3\underbrace{(s_0 - s_1)}_{1-h_1} - s_1 = -3 + 3h_1 - s_1 = 3(h_1 - 1) - s_1,
\]

\[
\dim M_G/\mathbb{Z} = 3\bigl(\underbrace{(\textstyle\sum g_\alpha) + h_1}_{g} - 1\bigr) + \nu - s_1
\]

\[
\text{(28)}\qquad \dim M_G/\mathbb{Z} = \underbrace{\bigl[3(g-1)+\nu\bigr]}_{= \dim \hat{M}_{g,\nu}/\mathbb{Z}} - s_1 ,
\]

\[
\text{(29)}\qquad \operatorname{codim}_{\hat{M}_{g,\nu}} M_{[G]} = s_1
\]

\begin{align*}
\text{(30)}\quad \mu(G) &= \sum_{\alpha\in S} \bigl(3(g_\alpha - 1) + \hat{\nu}_\alpha\bigr) = 3\sum g_\alpha - 3s_0 + \nu + 2s_1 \\
  &= \bigl(3(g-1) + \nu\bigr) - s_1
\end{align*}

\[
\hat{M}_G \overset{\text{déf}}{=} \prod_{\alpha\in S} \hat{M}_{g_\alpha, \hat{I}_\alpha}
\]

\[
\text{(30)}\qquad \mathcal{S} \longmapsto
\begin{array}{l}
\text{courbes de MD relatives } Y_\alpha/\mathcal{S}\text{,} \\
\text{de type } g_\alpha, \hat{I}_\alpha \text{ i.e.\ de genre } g_\alpha\text{, munies d'un mono} \\
(\hat{I}_\alpha)_{\mathcal{S}} \hookrightarrow Y_\alpha^{\text{lisse}}
\end{array}
\]

\[
\text{(31)}\qquad \mathcal{S} \longmapsto
\begin{array}{l}
\text{groupoïde des systèmes } (Y_\alpha)_{\alpha\in S} \text{ de courbes de MD } Y_\alpha \text{ de genre } g_\alpha\text{,} \\
\text{sur } \mathcal{S}\text{, munies de mono.\ } \hat{I}_{\alpha\,\mathcal{S}} \hookrightarrow Y_\alpha^{\text{lisse}} .
\end{array}
\]

\[
\text{(32)}\qquad
\begin{cases}
\hat{I} = \coprod \hat{I}_\alpha = I \sqcup \tilde{A} \\
Y = \coprod_{\alpha\in S} Y_\alpha
\end{cases}
\]

\[
\text{(33)}\qquad \hat{I}_{\mathcal{S}} \hookrightarrow Y^{\text{lisse}}
\]

\[
\text{(34)}\qquad \mathfrak{X} = Y/(\sigma_{\tilde{A}_{\mathcal{S}}}) ,
\]

\[
\text{(35)}\qquad I_{\mathcal{S}} \hookrightarrow \mathfrak{X}^{\text{lisse}}
\]

\[
\text{(36)}\qquad \hat{M}_G \longrightarrow \hat{M}_{g,I}
\]

\[
\text{(38)}\qquad G = (S, \tilde{A}, \sigma_{\tilde{A}}, I, \hat{I} \to S \xrightarrow{g} \mathbb{N}), \qquad \hat{I} = \tilde{A} \sqcup I
\]

\[
\hat{M}_G = \prod_{\alpha\in S} \hat{M}_{g_\alpha, \hat{I}_\alpha} .
\]

\[
\text{(39)}\qquad G(\alpha) = \bigl(S(\alpha), \tilde{A}(\alpha), \sigma_{\tilde{A}(\alpha)}, I(\alpha) = \hat{I}_\alpha, \ \hat{I}(\alpha) \overset{\text{déf}}{=} \tilde{A}(\alpha) \sqcup I(\alpha) \to S(\alpha) \xrightarrow{g_\alpha} \mathbb{N}\bigr)
\]

\[
\text{(40)}\qquad
\left\{
\begin{array}{l}
S' = \coprod_{\alpha\in S} S(\alpha) \\[2pt]
\tilde{A}' = \tilde{A} \sqcup \coprod_{\alpha\in S} \tilde{A}(\alpha) \\[2pt]
\sigma_{\tilde{A}'} \text{ induit par } \sigma_{\tilde{A}} \text{ et par les } \sigma_{\tilde{A}(\alpha)} \\[2pt]
I' = I \quad \text{donc} \quad \hat{I}' \overset{\text{déf}}{=} \tilde{A}' \sqcup I' \simeq \underbrace{(\tilde{A} \sqcup I)}_{\hat{I} = \coprod_{\alpha\in S} \hat{I}_\alpha = \coprod_{\alpha\in S} I(\alpha)} \sqcup \coprod_{\alpha\in S} \tilde{A}(\alpha) \\[2pt]
\phantom{I' = I \quad \text{donc} \quad \hat{I}'} \simeq \coprod_{\alpha\in S} \underbrace{\hat{I}_\alpha \sqcup \tilde{A}(\alpha)}_{\hat{I}(\alpha)} \\[2pt]
\hat{I}' \to S' \text{ induit par les } \hat{I}(\alpha) \to S(\alpha) \\[2pt]
S' \xrightarrow{g'} \mathbb{N} \text{ induit par les } S(\alpha) \xrightarrow{g_\alpha} \mathbb{N}
\end{array}
\right.
\]

\[
S(\alpha) \xrightarrow{g_\alpha} \mathbb{N}
\]

\[
\operatorname{card} I_{\alpha,\beta} + \operatorname{card} \tilde{A}_{\alpha,\beta} \geqslant 3 .
\]

\[
\text{(41)}\qquad f : G' \to G''
\]

\[
\text{(42)}\qquad \Pi_1(f) : \Pi_1(G') \to \Pi_1(G'') ,
\]

\[
\text{(44)}\qquad S' \sqcup \tilde{A}' \longrightarrow S'' \sqcup \tilde{A}''
\]

\[
\text{(46)}\qquad
\left|
\begin{array}{l}
S \xrightarrow{g} \mathbb{N}, \quad S' \xrightarrow{g'} \mathbb{N} \\
I \to S, \quad I' \to S'
\end{array}
\right.
\]

\[
\text{(47)}\qquad I' \simeq I'' ,
\]

\[
2g'_\beta + (\operatorname{card} I'_\beta + \operatorname{card} \tilde{B}'_\beta) \geqslant 3
\]

\[
\sum_{\beta\in S'_\alpha} g'_\beta + h_1(G(\alpha)) = g_\alpha .
\]

\[
\underbrace{\operatorname{card} I_\alpha}_{\nu_\alpha} = \sum_{\beta\in S'_\alpha} \underbrace{\operatorname{card} I'_\beta}_{\nu'_\beta}
\]

\[
g_\alpha = \Bigl(\sum_{\beta\in S'_\alpha} g'_\beta\Bigr) + h_1(G'(\alpha))
\]

\[
\pi_0(G'_{!}) \longrightarrow S
\]

\[
(49)\qquad S \simeq \pi_0(G'_{!}),
\]

\[
(50)\qquad \widetilde{A} \simeq \widetilde{B}'
\]

\[
(53)\qquad \widetilde{C}' = \widetilde{A}' \smallsetminus \widetilde{B}', \quad C' = A' \smallsetminus B' = \widetilde{C}'/\sigma_{\widetilde{A}'},
\]

\[
(54)\qquad G = G'/\widetilde{C}' \quad\text{ou}\quad G = G'/C'
\]

\[
(55)\qquad G' \to G_0 \to G,
\]

\[
I' \to S', \quad S' \xrightarrow{g'} \mathbb{N},
\]

\[
I \overset{\text{déf}}{=} I' \to S' \to S,
\]

\[
(56)\qquad g_\alpha = \Bigl(\sum_{\alpha' \in S'_\alpha} g'_{\alpha'}\Bigr) + h^1(G'(\alpha)).
\]

\[
2g_\alpha + \underbrace{\operatorname{card} I_\alpha + \operatorname{card} \widetilde{A}_\alpha}_{\hat{\nu}_\alpha} \geqslant 3,
\]

\[
I(\alpha) \overset{\text{déf}}{=} I'_\alpha \amalg \widetilde{B}'_\alpha \longrightarrow S'_\alpha
\]

\[
2g'_\beta + \underbrace{\bigl(\operatorname{card} I'_\beta + \operatorname{card} \widetilde{B}'_\beta\bigr)}_{\substack{\text{poids marqué}\\ \text{de } \beta \text{ dans } \underline{G}(\alpha)}} + \underbrace{\operatorname{card} \widetilde{C}'_\beta}_{\substack{\text{ordre de}\\ \beta \text{ dans } \underline{G}(\alpha)}} \geqslant 3
\]

\[
2g'_\beta + \underbrace{\operatorname{card} I'_\beta}_{\substack{\text{poids marqué}\\ \text{de } \beta \text{ dans } \underline{G}'_\beta}} + \underbrace{\operatorname{card} \widetilde{A}'_\beta}_{\substack{\text{ordre de } \beta\\ \text{dans } \underline{G}'}} \geqslant 3,
\]

\[
\underline{G}' \longrightarrow \underline{G}
\]

\[
\pi_0(G') \xrightarrow{\ \sim\ } \pi_0(G)
\]

\[
I'_\alpha \amalg \widetilde{B}'_\alpha
\]

\[
\underline{G} \longleftrightarrow B' = A' \smallsetminus C',
\]

\[
(57)\qquad \widehat{M}_{[\underline{G}]} \hookrightarrow \widehat{M}_{g,\nu}
\]

\[
(58)\qquad (\widehat{M}_G, \Gamma^{!}) \hookrightarrow \widehat{M}_{g,I}
\]

\[
(59)\qquad
\left\{
\begin{array}{l}
\Gamma^{!} \subset \Gamma = \operatorname{Aut}(\underline{G}) \\[2pt]
\Gamma^{!} \overset{\text{déf}}{=} \{\, u \in \Gamma \mid u_I = \mathrm{id}_I \,\} = \operatorname{Ker}\bigl(\operatorname{Aut}(\underline{G}) \to \mathfrak{S}_I\bigr),
\end{array}
\right.
\]

\[
(60)\qquad M_G = M_{\widehat{G}} \simeq \operatorname{Spec} \mathbb{Z} \overset{\text{déf}}{=} S_0,
\]

\[
(61)\qquad M_{[G]} \simeq B(\Gamma_{S_0})
\]

\[
(62)\qquad (M_{[G]}, \Gamma^{!}) \simeq B(\Gamma^{!}_{S_0})
\]

\[
(63)\qquad \operatorname{dimmod}(G) = 3(g-1) + \nu = 3\bigl(h_1(G) - 1\bigr) + \nu .
\]

\[
(64)\qquad
\left\{
\begin{array}{lll}
\bullet^{3} & \nu = 3, & \dim \operatorname{amb} G = 0 \\[4pt]
\overset{2}{\bullet}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \nu = 4, & \dim \operatorname{amb} G = 1 \\[4pt]
\overset{2}{\bullet}\!\!-\!\!\!-\!\!\overset{1}{\bullet}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \nu = 5, & \dim \operatorname{amb} G = 2
\end{array}
\right.
\]

\[
(65)\qquad
\left\{
\begin{array}{ll}
\text{boucle à un sommet, de poids } 1 & \nu = 1 = \dim \operatorname{amb} G \\[2pt]
\text{deux sommets de poids } 1 \text{ joints par deux arêtes} & \nu = 2 = \dim \operatorname{amb} G \\[2pt]
\text{boucle munie d'une queue, le bout de poids } 2 & \nu = 2 = \dim \operatorname{amb} G
\end{array}
\right.
\]

\[
(65)\qquad
\left\{
\begin{array}{lll}
\bullet^{3} & \Gamma \simeq \mathfrak{S}_3, \ \Gamma^{!} = \{1\} & (\dim \operatorname{amb} 0) \\[4pt]
\overset{2}{\bullet}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \Gamma \simeq D_4, \ \Gamma^{!} = \{1\} & (\dim \operatorname{amb} 1) \\[4pt]
\overset{2}{\bullet}\!\!-\!\!\!-\!\!\overset{1}{\bullet}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \Gamma \simeq D_4, \ \Gamma^{!} = \{1\} & (\dim \operatorname{amb} 2)
\end{array}
\right.
\]

\[
(66)\qquad
\left\{
\begin{array}{ll}
\text{boucle à un sommet de poids } 1 : & \Gamma = \Gamma^{!} \simeq \mathbb{Z}/2 \quad (\dim \operatorname{amb} 1) \\[2pt]
\text{deux sommets de poids } 1, \text{ deux arêtes} : & \Gamma \simeq D_2, \ \Gamma^{!} \simeq \mathbb{Z}/2 \quad (\dim \operatorname{amb} 2) \\[2pt]
\text{boucle avec queue, bout de poids } 2 : & \Gamma \simeq \mathbb{Z}/2 \times \mathbb{Z}/2, \ \Gamma^{!} \simeq \mathbb{Z}/2 \quad (\dim \operatorname{amb} 2)
\end{array}
\right.
\]

\[
(67)\qquad \delta(G) = \sum_{\alpha \in S} \underbrace{\bigl(3(g_\alpha - 1) + \nu_\alpha + \tilde{\mu}_\alpha\bigr)}_{\delta_\alpha}
\]

\[
\left\{
\begin{array}{l}
1 \text{ ou } 2 \text{ des } \delta_\alpha \text{ sont égaux à } 1, \text{ les autres sont nuls} \\
1 \text{ des } \delta_\alpha \text{ est égal à } 2, \text{ les autres sont nuls}
\end{array}
\right.
\]

\[
(68)\qquad
\left(
\begin{array}{l}
\alpha_0 \text{ de genre } 1, \text{ et son poids total } \nu_{\alpha_0} + \tilde{\mu}_{\alpha_0} \text{ est égal à } 1 \\
\quad \text{i.e.\ } \alpha_0 \text{ est d'ordre } 1, \text{ ou isolé de poids marqué } 1 \\
\underline{\text{ou}} \ \bigl(\alpha_0 \text{ de genre } 0, \text{ et son poids total } \nu_{\alpha_0} + \tilde{\mu}_{\alpha_0} = 4, \\
\quad \text{i.e.\ } (\nu_{\alpha_0}, \tilde{\mu}_{\alpha_0}) \in \{(4,0), (3,1), (2,2), (1,3), (0,4)\}\bigr)
\end{array}
\right)
\]

\[
(69)\qquad
\begin{array}{l}
\text{les sommets } \alpha \neq \alpha_0 \text{ sont de genre } 0, \text{ d'ordre } \tilde{\mu}_\alpha \leqslant 3, \\
\text{et pour un tel } \alpha \text{ le poids marqué } \nu_\alpha \text{ est égal à } 3 - \tilde{\mu}_\alpha .
\end{array}
\]

\[
(70)\qquad
\left\{
\begin{array}{ll}
\overset{4}{\bullet} \ \ g = 0 & \Gamma = \mathfrak{S}_4, \ \Gamma^{!} = 1 \\[6pt]
\overset{1}{\bullet} \ \ g = 1 & \Gamma = \Gamma^{!} = \{e\}
\end{array}
\right.
\]

\[
(71)\qquad \widehat{M}_G \simeq \widehat{M}_{g_{\alpha_0}, \hat{I}_{\alpha_0}}
\]

\[
(72)\qquad \Gamma_{\underline{G}} \longrightarrow \mathfrak{S}_{\hat{I}_{\alpha_0}} \quad (\simeq \mathfrak{S}_4)
\]

\[
(72)\qquad
\left\{
\begin{array}{lll}
\overset{3}{\underset{\alpha_0}{\bullet}}\!\!-\!\!\!-\!\!\overset{2}{\underset{\alpha}{\bullet}} & \text{i.e.\ deux droites } (a, b, c;\ d, e) & \text{type } (0,5) \\[6pt]
\text{boucle sur } \alpha_0, \text{ de poids } 2 & \text{i.e.\ courbe nodale } (a, b) & \text{type } (1,2) \\[6pt]
\overset{}{\underset{(\alpha_0),\ g_{\alpha_0} = 1}{\bullet}}\!\!-\!\!\!-\!\!\overset{2}{\underset{g_\alpha = 0}{\bullet}} & \text{i.e.\ genre } 1 + \text{droite } (a, b) & \text{type } (1,2)
\end{array}
\right.
\]

\[
(73)\qquad
\left\{
\begin{array}{ll}
\overset{3}{\underset{\alpha_0}{\bullet}}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \Gamma = \mathfrak{S}_3 \times \mathfrak{S}_2, \ \Gamma^{!} = 1, \ \operatorname{Im}(\Gamma \to \mathfrak{S}_4) \simeq \mathfrak{S}_3 \\[6pt]
\text{boucle sur } (\alpha_0), \text{ de poids } 2 & \Gamma \simeq \mathfrak{S}_2 \times \mathfrak{S}_2, \ \Gamma^{!} \simeq \mathfrak{S}_2, \ \operatorname{Im}(\Gamma \to \mathfrak{S}_4) \simeq \Gamma \\ & \qquad = \mathfrak{S}_2 \times \mathfrak{S}_2 \\[6pt]
\underset{g_{\alpha_0} = 1}{\bullet}\!\!-\!\!\!-\!\!\overset{2}{\bullet} & \Gamma \simeq \mathfrak{S}_2, \ \Gamma^{!} \simeq 1
\end{array}
\right.
\]

\[
(74)\qquad
\left\{
\begin{array}{lll}
\widehat{M}^{!}_{0,5} & \Gamma \text{ opère via} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0}} \ (\simeq \mathfrak{S}_5) \\[2pt]
\widehat{M}^{!}_{1,2} & \Gamma \text{ ------} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0}} \simeq \mathbb{Z}/2\mathbb{Z} \\[2pt]
\widehat{M}^{!}_{0,4} \times \widehat{M}^{!}_{0,4} & \Gamma \text{ ------} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0} \amalg \hat{I}_{\alpha_1}} \simeq \mathfrak{S}_8 \\[2pt]
\widehat{M}^{!}_{0,4} \times \widehat{M}^{!}_{1,1} & \Gamma \text{ ------} & \Gamma \to \mathfrak{S}_{\hat{I}_{\alpha_0}} \simeq \mathfrak{S}_4 \\[2pt]
\widehat{M}^{!}_{1,1} \times \widehat{M}^{!}_{1,1} & \Gamma \text{ opère trivialement.} &
\end{array}
\right.
\]

\[
\Gamma \longrightarrow \underbrace{\mathbb{Z}/2 \cdot (\mathfrak{S}_4 \times \mathfrak{S}_4)}_{\text{prod.\ }\tfrac{1}{2}\text{ direct}} .
\]

\[
(75)\qquad
\left\{
\begin{array}{ll}
M^{!}_{0,4} & \Gamma \text{ opère via } \Gamma \to \mathfrak{S}_{I_{\alpha_0}} \simeq \mathfrak{S}_4 \\[2pt]
\widehat{M}^{!}_{1,1} & \Gamma \text{ opère trivialement,}
\end{array}
\right.
\]

\[
(76)\qquad
\begin{array}{c|c}
\text{Codim } 0 & \text{Codim } 1 \\ \hline
\overset{I(4)}{\bullet} \quad \xrightarrow{\ 3\ } & \overset{I'(2)}{\bullet}\!\!-\!\!\!-\!\!\overset{I''(2)}{\bullet}
\end{array}
\]

\[
(77)\qquad \widehat{M}^{!}_{0,4} \simeq \mathbb{P}^1_{\mathbb{Z}}, \quad M^{!}_{0,4} \simeq (U_{0,3})_{\mathbb{Z}} = \mathbb{P}^1_{\mathbb{Z}} \smallsetminus \{0, 1, \infty\}
\]

\[
(78)\qquad
\begin{array}{c|c}
\text{codim } 0 & \text{codim } 1 \\ \hline
\overset{4}{\bullet} \;\xrightarrow{\;1\;} & \overset{2}{\bullet}\!\text{---}\!\overset{2}{\bullet}
\end{array}
\]

\begin{align*}
\chi_!(M^{!\,\mathrm{an}}_{0,4}) &= \chi_!(U^{\mathrm{an}}_{0,3}) = 2 - 2\underset{0}{g} - \underset{3}{\nu} = -1 \\
\chi_!(\widehat{M}^{!\,\mathrm{an}}_{0,4}) &= \chi_!(\mathbb{P}^{1\,\mathrm{an}}) = 2 \\
\chi_!(M^{\mathrm{an}}_{0,4}) &= \chi_!(\mathbb{B}^{\mathrm{an}}_{0,3}, \mathfrak{S}_4)
= \chi_!\bigl(U_{0,3} \setminus \{-j, -\bar{j}\} \setminus \{\tfrac12, 2, -1\}, \mathfrak{S}_4\bigr) \\
&\qquad + \chi_!(\{-j, -\bar{j}\}, \mathfrak{S}_4) + \chi_!(\{\tfrac12, 2, -1\}, \mathfrak{S}_4) \\
&= -1 + \tfrac13 + \tfrac12 = -\tfrac{1}{24}
\end{align*}

\[
\chi_!(\widehat{M}^{\mathrm{an}}_{0,4}) = \underbrace{\chi_!(U^{\mathrm{an}}_{0,3}, \mathfrak{S}_4)}_{-1/24} + \underbrace{\chi_!(\{0, 1, \infty\}, \mathfrak{S}_4)}_{1/8} = \frac{1}{12} .
\]

\begin{align*}
[M^{!}_{0,4}] &= [U_{0,3}] = \underbrace{T(1)}_{\text{Tate} \simeq [\mathbb{E}^1]} - 2 \\
[M_{0,4}] &= \tfrac14\bigl(T(1) - 2\bigr) + \tfrac{1}{12} + \tfrac18 = \tfrac14\bigl(T(1) - 1\bigr) - \tfrac{1}{24} \\
[\widehat{M}_{0,4}] &= [M_{0,4}] + \tfrac18 = \tfrac14\bigl(T(1) - 1\bigr) + \tfrac{1}{12}
\end{align*}

\[
(77)\qquad
\begin{array}{c|c}
\text{codim } 0 & \text{codim } 1 \\ \hline
\underset{1}{\overset{1}{\bullet}} \;\xrightarrow{\;1\;} & \text{(boucle)}\overset{1}{\bullet}
\end{array}
\]

\[
(78)\qquad \widehat{M}_{1,1} \longrightarrow \widehat{M}'_{1,4} \overset{\mathrm{def}}{=} (\widehat{M}^{!}_{1,4}, \mathfrak{S}_3)
\]

\[
[M_{1,1}] = \tfrac12 [U_{0,3}, \mathfrak{S}_3] = \tfrac12 [M_{0,4}] \text{\struck{$\ldots$}}
\]

\[
(79)\quad
\begin{cases}
[M_{1,1}] = \tfrac12\bigl(T(1) - 1\bigr) - \tfrac{1}{12} \\
[\widehat{M}_{1,1}] = [M_{1,1}] + \underset{1/2}{[\widehat{M}^{\infty}_{1,1}]} = \tfrac12\bigl(T(1) - 1\bigr) + \tfrac{5}{12}
\end{cases}
\]

\[
M_{1,1} \longrightarrow \mathbb{E}^1
\]

\[
\mathbb{E}^1 = U \amalg S'_0 \amalg S'_1 \cup a_2 \cup a_3
\]

\[
\begin{cases}
U = \mathbb{E}^1 \setminus (S_0 \cup S_1) \\
\{a_2, a_3\} = S_0 \cap S_1 \quad (a_2 \text{ en car. } 2,\ a_3 \text{ en car. } 3) \\
S'_0 = S_0 \setminus \{a_2, a_3\},\quad S'_1 = S_1 \setminus \{a_2, a_3\}
\end{cases}
\]

\[
[M_{1,1}] = \tfrac12 [U] + \tfrac14 [S'_0] + \tfrac16 [S'_1] + \tfrac{1}{24} [a_2] + \tfrac{1}{12} [a_3]
\]

\begin{align*}
[U] &= [\mathbb{E}^1] - [S_0] - [S_1] + [a_2] + [a_3] \\
&= T(1) - 2 + \zeta_2 + \zeta_3
\end{align*}

\[
\zeta_p = [\operatorname{Spec} \mathbb{Z}/p\mathbb{Z}] = i_{p!}(1)
\]

\begin{align*}
[S'_0] &= [S_0] - [a_2] - [a_3] = 1 - \zeta_2 - \zeta_3 \\
[S'_1] &= [S_1] - [a_2] - [a_3] = 1 - \zeta_2 - \zeta_3
\end{align*}

\begin{align*}
[M_{1,1}] &= \tfrac12\bigl(T(1) - 2 + \zeta_2 + \zeta_3\bigr) + \underbrace{\bigl(\tfrac14 + \tfrac16\bigr)}_{5/12}\bigl(1 - \zeta_2 - \zeta_3\bigr) \\
&\qquad + \tfrac{1}{24}\zeta_2 + \tfrac{1}{12}\zeta_3 \\
&= \tfrac12 T(1) - \tfrac{7}{12} + \tfrac18 \zeta_2 + \tfrac16 \zeta_3 \\
&= \tfrac12\bigl(T(1) - 1\bigr) - \tfrac{1}{12} + \tfrac18 \zeta_2 + \tfrac16 \zeta_3 ,
\end{align*}

\[
(80)\quad
\begin{cases}
[M_{1,1}] = \tfrac12\bigl(T(1) - 1\bigr) - \tfrac{1}{12} + \tfrac18 \zeta_2 + \tfrac16 \zeta_3 \\
[\widehat{M}_{1,1}] = [M_{1,1}] + \tfrac12 = \tfrac12 T(1) - \tfrac{1}{12} + \tfrac18 \zeta_2 + \tfrac16 \zeta_3
\end{cases}
\]

\[
(81)\qquad
\begin{array}{c|c|c}
\text{codim } 0 & \text{codim } 1 & \text{codim } 2 \\ \hline
\overset{I(5)}{\bullet} \;\xrightarrow{\;(10)\;} &
\overset{I'(2)}{\bullet}\!\text{---}\!\overset{I''(3)}{\bullet} \;\xrightarrow{\;(3)\;} &
\overset{I'(2)}{\bullet}\!\text{---}\!\overset{I'''(1)}{\bullet}\!\text{---}\!\overset{I''(2)}{\bullet}
\end{array}
\]

\[
(82)\qquad \widehat{M}_{0,5} = (\widehat{M}^{!}_{0,5}, \mathfrak{S}_5) = (\widehat{M}_{0,I}, \mathfrak{S}_I)
\]

\[
(83)\qquad
\begin{array}{c|c|c}
\text{codim } 0 & \text{codim } 1 & \text{codim } 2 \\ \hline
\underset{1}{\overset{I(2)}{\bullet}} &
\begin{array}{l} G_0 = \text{(boucle sur un sommet } I(2)\text{)} \\ G_1 = \overset{1}{\bullet}\!\text{---}\!\overset{I(2)}{\bullet} \end{array} &
\begin{array}{l} \overset{\{a\}}{\bullet}\!=\!\overset{\{b\}}{\bullet} = P_0 \\ \text{(boucle)}\!\bullet\!\text{---}\!\overset{I(2)}{\bullet} = P_1 \end{array}
\end{array}
\]

\[
(84)\qquad J = \{\varepsilon_1\} \amalg J' \qquad J' = I \wedge_{\mathbb{Z}/2} \widetilde{A}^{!}_0 .
\]

\[
(85)\qquad
\begin{aligned}
\widehat{M}_{G_0} &\simeq \bigl(\mathbb{P}^1_{\{\varepsilon_1\} \amalg J'}\text{\struck{$, \mathfrak{S}_{J'}$}}\bigr) \\
\underbrace{(\widehat{M}_{G_0}, \Gamma^{!}_0)}_{\widehat{M}_{D_{\circ}}} &\simeq \bigl(\mathbb{P}^1_{(\varepsilon_1 \amalg J')}, \mathfrak{S}_{J'}\bigr)
\end{aligned}
\]

\[
(86)\qquad M_{D_{\circ}} \simeq \mathbb{E}^1_{\mathbb{Z}} \setminus \{\text{section } 2\} = \operatorname{Spec} \mathbb{Z}[T]_{(T-2)}
\]

\[
\text{via}\quad
\begin{cases}
M_{G_0} \longrightarrow M_{D_{\circ}} \\
T \longmapsto T + \frac{1}{T}
\end{cases}
\]

\[
(87)\qquad \widehat{M}_{D_{\circ}} \simeq (\mathbb{P}^1_{\mathbb{Z}}, \text{avec singularité } 2 \text{ le long de la section } T = 2)
\]

\[
(88)\qquad \widehat{M}_{D_1} \simeq \widehat{M}_{G_1} \simeq \widehat{M}_{1,1} \simeq (\mathbb{P}^1_{\mathbb{Z}}, \ldots)
\]

\[
(89)\qquad
\begin{cases}
\widehat{M}_G \simeq \prod\limits_{\alpha \in S} \widehat{M}^{!}_{g_\alpha, \hat{\nu}_\alpha} \\
M_G \simeq \prod\limits_{\alpha \in S} M^{!}_{g_\alpha, \nu_\alpha}
\end{cases}
\]

\[
\widehat{M}_G \simeq \widehat{M}^{!}_{g,\nu} \quad (\simeq M_{g,I} \ldots)
\]

\[
(90)\quad
\begin{cases}
\text{a)}\ \ \text{une boucle sur un sommet } \overset{I}{\underset{g-1}{\bullet}} \quad (\text{si } g \geq 1) \\
\text{b)}\ \ \overset{I'}{\underset{g'}{\bullet}}\!\text{---}\!\overset{I''}{\underset{g''}{\bullet}} \quad
\begin{cases}
I = I' \amalg I'' \\
g' + g'' = 0 \\
\text{si } g' \text{ (ou } g'') = 0, \text{ alors} \\
\quad \text{\struck{$2g' + \operatorname{card} I' \geq 2$}} \\
\quad \text{\struck{$2g'' + \operatorname{card} I'' \geq 2$}} \\
\quad \operatorname{card} I' \geq 2 \ (\text{ou } \operatorname{card} I'' \geq 2)
\end{cases}
\end{cases}
\]

\[
(91)\quad \text{a}')\quad
\text{une boucle sur } \overset{I_1}{\underset{g_1}{\bullet}}\!\text{---}\!\overset{I_2}{\underset{g_2}{\bullet}}
\qquad
\begin{cases}
I_1 \amalg I_2 = I \\
g_1 + g_2 = g - 1 \\
\text{si } g_2 = 0 \text{ \struck{\ill{}}}\ \text{\add{on a card } } I_2 \geq 2 \\
\quad \text{\struck{($I_1 \neq \emptyset$ et card $I_2 \neq \emptyset$)}}
\end{cases}
\]

\[
(92)\quad \text{b}')\quad
\text{une boucle sur } \overset{I'}{\underset{g'-1}{\bullet}}\!\text{---}\!\overset{I''}{\underset{g''}{\bullet}}
\]

\[
\text{b}'')\qquad
\text{\struck{(b'') : \ill{}}}\quad
\overset{\mathfrak{f}}{\bullet}\!\text{---}\!\bullet
\]

\[
(93)\quad \text{b}'')\quad
\underbrace{\overset{2}{\underset{(a_0)}{\bullet}}\!-\!\overset{1}{\underset{(a_1)}{\bullet}}\!-\!\overset{1}{\bullet}\cdots\overset{1}{\underset{(a_{k-1})}{\bullet}}\!-\!\overset{1}{\underset{a_k}{\bullet}}}_{\substack{k \text{ segments},\ k+1 \text{ sommets} \\ \text{ayant un poids marqué total} \\ 2 + k = \operatorname{card} I'}}
\!-\!
\underbrace{\overset{1}{\underset{(b_l)}{\bullet}}\!-\!\overset{1}{\underset{(b_{l-1})}{\bullet}}\cdots\overset{1}{\underset{(b_2)}{\bullet}}\!-\!\overset{1}{\underset{(b_1)}{\bullet}}\!-\!\overset{2}{\underset{(b_0)}{\bullet}}}_{\substack{l \text{ segments},\ l+1 \text{ sommets} \\ \text{poids marqué total} \\ 2 + l = \operatorname{card} I''}}
\]

\[
\overset{(c_1, c_2)}{\underset{c}{\bullet}}\!-\!\overset{c_3}{\bullet}\!-\!\cdots\!-\!\overset{c_{i-1}}{\bullet}\!-\!\overset{(c_i, c_{i+1})}{\bullet}\!-\!\overset{c_{i+2}}{\bullet}\!-\!\cdots\!-\!\overset{c_\nu}{\bullet}
\]

\[
\cdots\!-\!\overset{c_{i-1}}{\bullet}\!-\!\overset{c_i}{\bullet}\!-\!\overset{c_{i+1}}{\bullet}\!-\!\overset{c_{i+2}}{\bullet}\!-\!\cdots
\]

\[
\cdots\!-\!\overset{c_{i-1}}{\bullet}\!-\!\overset{c_{i+1}}{\bullet}\!-\!\overset{c_i}{\bullet}\!-\!\overset{c_{i+2}}{\bullet}\!-\!\cdots
\]

\[
(94)\qquad \Gamma_{\underline{G}} = \operatorname{Aut} \underline{G}
\]

\[
(95)\qquad \pi_1(\Pi M_{\underline{G}}) = \pi_1(M_{\underline{G}}) \simeq \prod_{\alpha \in S(\underline{G})} \pi_1(M_{g_\alpha, \hat{I}_\alpha}) \simeq \prod_\alpha \Gamma^{+}_{g_\alpha, I_\alpha} \simeq \prod_\alpha \mathcal{T}^{!+}_{g_\alpha, \nu_\alpha}
\]

\[
(96)\qquad 1 \to \pi_1(\Pi M_{\underline{G}}, \Gamma_{\underline{G}}) \to \pi_1(\Pi M_{\underline{G}}, \Gamma_{\underline{G}}) \longrightarrow \Gamma_{\underline{G}} \to 1
\]

\[
(96)\qquad \underline{G}' = \underline{G} / C .
\]

\[
(98)\qquad (\widehat{M}_{\underline{G}}, \Gamma^{!}_{\underline{G},\underline{G}'}) \hookrightarrow \widehat{M}_{\underline{G}'}
\]

\[
(99)\qquad (M_{\underline{G}}, \Gamma^{!}_{\underline{G},\underline{G}'}) \hookrightarrow \widehat{M}_{\underline{G}'}
\]

\[
(100)\qquad \Pi M^{*}_{\underline{G},\underline{G}'}, \quad \Pi M^{*}_{\underline{G},\underline{G}'}
\]

\[
(101)\qquad
\begin{array}{ccc}
M^{*}_{\underline{G},\underline{G}'} & \longrightarrow & M_{\underline{G}'} \\
\big\downarrow & & \\
(M_{\underline{G}}) \overset{\text{\struck{\ill{}}}}{\longrightarrow} M_{\underline{G},\underline{G}'} & &
\end{array}
\]

\[
(102)\qquad
\begin{array}{ccc}
\Pi M^{*}_{\underline{G},\underline{G}'} & \longrightarrow & \Pi M_{\underline{G}'} \\
\big\downarrow & & \\
\Pi(M_{\underline{G}}) \overset{\approx}{\longrightarrow} \Pi M_{\underline{G},\underline{G}'} & &
\end{array}
\]

\[
(103)\qquad
\begin{array}{ccc}
\Pi M^{*}_{\underline{G},\underline{G}'} & \xrightarrow{\ \varphi_{\underline{G},\underline{G}'}\ } & \Pi M_{\underline{G}'} \\
\big\downarrow{\scriptstyle \varphi_{\underline{G},\underline{G}'}} & & \\
\Pi(M_{\underline{G}}, \ldots) & &
\end{array}
\]

\[
(104)\qquad \underline{G} \longrightarrow \underline{G}' \longrightarrow \underline{G}''
\]

\[
\text{\struck{$\Pi M^{*}_{\underline{G},\underline{G}'} \longrightarrow \ldots$ \quad $\big\downarrow$ \quad $\Pi M_{\underline{G}}$}}
\]

\[
\Pi M_{G'} \;\bigl(\simeq \Pi(M_{G'},\Gamma^{!}_{G',G''})\bigr)
\]

\[
(106)\qquad \Pi M^{*}_{G,G',G''}
\]

\[
(109)\qquad \text{\struck{$\Pi M_{G,G'} \to \Pi M_{G,G''}$,}}
\]

\[
(110)\qquad \text{\struck{$\Gamma^{!}_{G,G'} \subset \Gamma^{!}_{G,G''}$}}
\]

\[
(111)\qquad \text{\struck{$\Pi(M_{G},\Gamma^{!}_{G,G'}) \xrightarrow{\ \mathrm{can}\ } \Pi(M_{G},\Gamma^{!}_{G,G''})$}}
\]

\[
\text{\struck{$\Pi M_{G,G'} \to \Pi M_{G,G''}$.}}
\]

\[
(113)\qquad S_{A}\Pi M_{G} \overset{\mathrm{déf}}{=} \Pi M^{*}_{G,G'} ,
\]

\[
(114)\qquad S_{A}\Pi M_{G} \longrightarrow (\Pi M_{G},\ \cdot\,)
\]

\[
\Gamma' \subset \Gamma_{G}
\]

\[
(115)\qquad (S_{A}\Pi M_{G},\Gamma') \longrightarrow (\Pi M_{G},\Gamma')
\]

\[
\simeq S_{A}\Pi(M_{G},\Gamma') \longrightarrow \Pi(M_{G},\Gamma')
\]

\[
(116)\qquad \mathbb{Z}^{A(G)} \longrightarrow \mathbb{Z}^{C}
\]

\[
(117)\qquad (S_{C}\Pi M_{G},\Gamma') .
\]

\[
(118)\qquad \text{\struck{$(S_{C}M_{G},\Gamma_{G,G/C}) \simeq \Pi M^{*}_{G,G/C}$}}
\]

\[
(119)\qquad S_{A}\Pi M_{G} \longrightarrow S_{A'}\Pi M_{G'}
\]

\[
(121)\qquad (S_{C}\Pi M_{G},\Gamma^{!}_{G,G'}) \longrightarrow \Pi M_{G'} ,
\]

\[
(122)\qquad (S_{C}\Pi M_{G},\Gamma') \longrightarrow (\Pi M_{G'},\Gamma'/\Gamma^{!}_{G,G'})
\]

\[
\simeq (\Pi M_{G,G'},\Gamma'/\Gamma^{!}_{G,G'})
\]

\[
(124)\qquad S\Pi M_{G} \xrightarrow{\ \varphi_{G}\ } \Pi M_{G} .
\]

\[
S\Pi M_{G} \xrightarrow{\ S\Pi(f)\ } S\Pi M_{G'} \xrightarrow{\ \varphi_{G'}\ } \Pi M_{G'}
\]

\[
M_{G} \longrightarrow M_{g_{\alpha},\hat{I}_{\alpha}}
\]

\[
M^{\natural}_{g,I} \longrightarrow M_{g,I'} \quad \text{pour } I' \subset I .
\]

\[
(126)\qquad I' \longrightarrow S \quad \text{induite par } I \to S \text{ et } \tilde{A}(G) \xrightarrow{\ \text{origine}\ } S ,
\]

\[
I \amalg \tilde{J} \longrightarrow S \quad \text{induite par } I \to S \text{ et par } \tilde{A} \xrightarrow{\ o_{G}\ } S .
\]

\[
I' = I_{\alpha} \amalg \tilde{A}_{\alpha} = \text{image inverse de } \{\alpha\} \text{ par } I \amalg \tilde{A} \to S .
\]

\[
\underline{G} = (\underbrace{S,\tilde{A},\sigma_{\tilde{A}},o_{G}}_{G},\ I \to S \xrightarrow{\ g\ } \mathbb{N})
\quad\text{et}\quad
\underline{G}' = (\underbrace{S',\dots}_{G'})
\]

\[
(130)\qquad G' \longrightarrow G .
\]

\[
(131)\qquad M_{G}(S) \longrightarrow M_{G'}(S) ,
\]

\[
(132)\qquad M_{G} \longrightarrow M_{G'} ,
\]

\[
(133)\qquad \text{\struck{$\Pi_{1}M_{G}$, $\Pi_{1}$}}\ M_{G} \longrightarrow \prod_{\alpha} M_{G_{\alpha}}
\]

\[
  \text{(144)}\qquad \Pi_1 M_G \;\simeq\; \prod_{\alpha\in S} \Pi_1 M_{g_\alpha, \hat{I}_\alpha}
\]

\[
  \text{(147)}\qquad \Pi_1 M_{g_\alpha, \hat{I}_\alpha} \longrightarrow \Pi_1 M_{g'_\alpha, \hat{I}'_\alpha}
\]

\[
  \text{(148)}\qquad \Pi_1 M_{g,I} \longrightarrow \Pi_1 M_{g,I'} \qquad \text{pour } I' \subset I,
\]

\[
  \mathfrak{S}_{I,I'} \simeq \mathfrak{S}_{I'} \times \mathfrak{S}_{I\setminus I'} \;\ldots
\]

\[
  \Pi_1 M_G \simeq \prod_{\alpha\in S(G)} \Pi_1 M_{\alpha, \hat{I}_\alpha} \quad\text{par}\quad \mathbb{Z}^{\tilde{A}(G)},
\]

\[
  \text{(149)}\qquad \text{\struck{$G \to G_{g,I}$,}}
  \qquad \text{\struck{i.e.\ un iso.\ $I(G) \simeq I$}}
\]

\[
  \text{(149)}\qquad \operatorname{Codim}(D_{x,J}, X_x) = \operatorname{card} J
\]

\[
  \text{(150)}\qquad J \overset{\varphi}{\hookrightarrow} \Delta_x \simeq \mathfrak{P}(I(x))^{\circ}
\]

\[
  \text{(151)}\qquad \text{si } x < y \text{ alors } c(x) < c(y),
\]

\[
  D_d \longrightarrow X
\]

\[
  \widetilde{D}_d \longrightarrow D_1, \qquad (x, V, W) \longmapsto W .
\]

\[
  \text{(152)}\qquad J \overset{\varphi}{\hookrightarrow} \Delta_x \simeq \mathfrak{P}(I(x))^{\circ}
\]

\[
  \text{(153)}\qquad D_J \longrightarrow X
\]

\[
  \text{(154)}\qquad H(J) = \text{ensemble des plongements d'ens.\ ordonnés,}
\]

\[
  \text{(155)}\qquad D_J \simeq D^{!}_d \overset{(\mathfrak{S}_d)_{D_d}}{\wedge} H(J) .
\]

\[
  \text{(156)}\qquad J' \overset{f}{\longrightarrow} J
\]

\[
  \text{(157)}\qquad D_J \longrightarrow D_{J'} ,
\]

\[
  \text{(158)}\qquad D_J \longrightarrow D_{J'} \overset{f_x}{\longrightarrow} D_\delta
\]

\[
  \text{(159)}\qquad D_J \longrightarrow D_{d,\delta}
\]

\[
  \text{(160)}\qquad D_{d,\delta} \simeq D^{!}_d \overset{(\mathfrak{S}_d)_{D_d}}{\wedge} \mathfrak{P}_\delta(I_d)
\]

\[
  \text{(161)}\qquad H(J) \longrightarrow \mathfrak{P}_\delta(I_d), \qquad h \longmapsto h(x)
\]

\[
  \text{\struck{(162)\quad $D_{d,\delta} \to D_{\delta'}$}}
\]

\[
  \text{(164)}\qquad D_{d,\delta} \longrightarrow D_\delta \qquad (d > \delta)
\]

\[
  \text{(165)}\qquad D_{d,d'} \times_{D_{d'}} D_{d',d''} \simeq D_{d,d',d''}
  \qquad \text{si } d > d' > d''
\]

\[
  \text{(167)}\qquad D_{d,d',d''} \longrightarrow D_{d,d''},
\]

\[
  \text{(168)}\qquad D_{d,d''} \longrightarrow D_d \times D_{d''}
\]

\[
  D_{d,d',d''} \rightrightarrows D_d,\ D_{d''}
\]

\[
  \text{(169)}\quad
  \begin{cases}
    D_{*} & \text{espace analytique des objets} \\
    D_{**} & \text{\phantom{espace analytique des} flèches}
  \end{cases}
\]

\[
  D_{*} \longrightarrow D_{**} \qquad \text{morphisme « identité »} ;
\]

\[
  (D_{**}, b) \times_{D_{*}} (D_{**}, s) \longrightarrow D_{**}, \qquad (u, v) \longmapsto v \circ u
\]

\[
  \text{(170)}\qquad D_{*} \overset{c}{\longrightarrow} \mathbb{N}
\]

\[
  \text{(171)}\qquad D_{*} = \coprod_{d\in\mathbb{N}} D_d
\]

\[
  \text{(172)}\qquad D_{**} = \coprod_{(d,d')\in\mathbb{N}\times\mathbb{N}} D_{d,d'}
\]

\[
  \text{(173)}\qquad D_{d,d'} = \text{image inverse de } D_d \times D_{d'}
  \text{ par } D_{**} \xrightarrow{(s,b)} D_{*} \times D_{*} .
\]

\[
    D_d \simeq D_{d,d}
  \]

\[
  \Bigl(\; D_{d,*} \overset{s_d}{\longrightarrow} D_d, \qquad
  D_{d,*} \overset{\text{déf}}{=} \coprod_{d'\in\mathbb{N}} D_{d,d'} = s^{-1}(D_d) \;\Bigr)
\]

\[
  D_{d,d'} \times_{D_{d'}} D_{d',d''} \longrightarrow D_{d,d''}
\]

\[
  D_{d,d''} \longrightarrow D_d \times D_{d''}
\]

\[
    D_{d,d'} \overset{b}{\longrightarrow} D_{d'} \qquad (\text{pour } 0 \leq d' < d),
  \]

\[
  \text{\struck{$D_{d,d',d''} \overset{\text{déf}}{=} D_{d,d'} \times_{D_{d'}} D_{d',d''} \to D_d \times D_{d''}$}}
\]

\[
  D_{d,d'} \simeq D^{!}_d \overset{(\mathfrak{S}_d)_{D_d}}{\wedge} \mathfrak{P}_{d'}(I_d),
\]

\[
  D_{d,d',d''} \overset{\text{déf}}{=} D^{!}_{d,d'} \overset{(\mathfrak{S}_{d'})_{D_{d,d'}}}{\wedge} \mathfrak{P}_{d''}(I_{d'}),
\]

\[
  D_{d,d'} \longrightarrow D_{d'}
\]

\[
  \text{(180)}\qquad D_d \longrightarrow D_0 \text{ est de codim.\ } d
\]

\[
  \text{(181)}\qquad (D^{!}_d)_x \simeq \mathfrak{P}(I_x) \;)
\]

\[
  \text{(18\ill{})}\qquad I_x = \text{\uncertain{ensemble} des sections en $x$ du rev.\ $D_{d,1}$ au-dessus de $D_d$}
\]

\[
  \simeq (D_{d,1})_x \;\text{---},
\]

\[
  D_d \overset{s_i}{\longrightarrow} D_{d,1} \overset{\mathrm{can}}{\longrightarrow} D_1 \longrightarrow X
\]

\[
  X = D_0 \simeq E^2, \qquad D_1 = \text{\struck{\ill{}}\ \emph{lisse} conn.}, \qquad D_2 = \emptyset \;\ldots
\]

\[
  \text{(183)}\qquad T_d = D^{!}_d \overset{(\mathfrak{S}_d)_{D_d}}{\wedge} (\mathbb{Z}^d)_{D_d} ,
\]

\[
  \text{(184)}\qquad G_d = T_d \otimes_{\mathbb{Z}} \mathbb{G}_{m, D_d}
  \simeq D^{!}_d \overset{(\mathfrak{S}_d)_{D_d}}{\wedge} \mathbb{G}_m^d
\]

\[
  (185) \qquad s_i : D^{!}_d \longrightarrow D_1 \qquad (1 \leqslant i \leqslant d)
\]

\[
  (186) \qquad L_1 \ \text{faisceau inversible canonique sur } D_1
\]

\[
  (187) \qquad L_{d,i} = s_i^{*}(L_1) \qquad 1 \leqslant i \leqslant d .
\]

\[
  (188) \qquad \underline{L}^{*}_{d,i} = \mathbb{G}_{m,D_d}\text{-torseur sur } D_d
  \ \text{associé au faisceau inv. } L_{d,i}
\]

\[
  (189) \qquad P^{!}_d = \prod_{1 \leqslant i \leqslant d} \underline{L}^{*}_{d,i}
  \qquad \text{torseur sous } \mathbb{G}_m^{d} .
\]

\[
  (190) \qquad G^{!}_d \overset{\text{déf}}{=} \mathbb{G}_{m,D^{!}_d} \otimes_{\mathbb{Z}} T^{!}_d
  \simeq \underbrace{\mathbb{G}^{d}_{m,D^{!}_d}}
\]

\[
  (191) \qquad P_d \ \text{torseur sur } D_d, \text{ de groupe } G_d ,
\]

\[
  (192) \qquad T_d \simeq \text{syst. local des } H_1 \text{ (ou des } \pi_1)
  \text{ des fibres de } P_d .
\]

\[
  \Pi\bigl(\underbrace{P_d \mid D^{*}_d}_{\overset{\text{déf}}{=} P^{*}_d}\bigr) = \Pi P^{*}_d .
\]

\[
  d \geqslant d' \geqslant 0
\]

\[
  (193) \quad
  \begin{cases}
    T(d,d')_d = (T_d)_{D_{d,d'}} & \text{syst. local de rang } d \text{ sur } D_{d,d'} \\
    T(d,d')_{d'} = (T_{d'})_{D_{d,d'}} & \text{syst. local de rang } d' \text{ sur } D_{d,d'}
  \end{cases}
\]

\[
  (194) \qquad T(d,d')_d \longrightarrow T(d,d')_{d'}
\]

\[
  (196) \qquad \mathbb{Z}^{d} \longrightarrow \mathbb{Z}^{d'}
\]

\[
  (197) \quad
  \begin{array}{ccc}
    G(d,d')_d & \longrightarrow & G(d,d')_{d'} \\
    \| \text{déf} & & \| \text{déf} \\
    G_d \times_{D_d} D_{d,d'} & & G_{d'} \times_{D_{d'}} D_{d,d'} \\
    \simeq T(d,d')_d \otimes_{\mathbb{Z}} \mathbb{G}_m & & \simeq T(d,d')_{d'} \otimes_{\mathbb{Z}} \mathbb{G}_m
  \end{array}
\]

\[
  (198) \quad
  \begin{array}{ccc}
    P(d,d')_{d'} & \simeq & \underbrace{P(d,d')_d} \wedge^{G(d,d')_d} G(d,d')_{d'} \\
    \| \text{déf} & & \| \text{déf} \\
    P_d \times_{D_d} D_{d,d'} & & P_{d'} \times_{D_{d'}} D_{d,d'}
  \end{array}
\]

\[
  d \geqslant d' \geqslant d''
\]

\[
  (199) \qquad T(d,d',d'')_d,\ T(d,d',d'')_{d'},\ T(d,d',d'')_{d''},
\]

\[
  (200) \qquad G(d,d',d'')_d,\ G(d,d',d'')_{d'},\ G(d,d',d'')_{d''}
\]

\[
  (201) \qquad P(d,d',d'')_d,\ P(d,d',d'')_{d'},\ P(d,d',d'')_{d''},
\]

\[
  \text{\struck{$(203) \quad \Pi_1 P^{*}_d \xrightarrow{\ r_{d,d'}\ } \Pi_1 P^{*}_{d'}$
  \quad pour $d' < d$}}
\]

\[
  (203) \qquad \Delta_r = [0,r] \cap \mathbb{N}
\]

\[
  (204) \qquad \mathcal{D}_r = \text{\struck{$\mathbf{D}$}}\, D_{\Delta_r}
  = \coprod_{\substack{(d_0,\dots,d_r) \in \mathbb{N}^{r} \\ d_0 > d_1 > \dots > d_r}}
  D_{d_0,d_1,\dots,d_r}
\]

\[
  (205) \qquad D_{d_*} = D^{!}_{d_0} \wedge^{(\mathfrak{S}_{d_0})_{D_{d_0}}}
  \underbrace{F_{d_*}(\text{\struck{$[1,d_0]$}}\, I_{d_0})}
\]

\[
  (206) \quad
  \begin{aligned}
    \mathcal{D}^{*}_{*} &= \coprod_{d_0 \geqslant d_1 \geqslant \dots \geqslant d_r \geqslant 0} D^{*}_{d_0,d_1,\dots,d_r} \\
    &= \text{image inverse de } D^{*}_{*} = \coprod_{d \in \mathbb{N}} D^{*}_d
    \text{ par le morphisme } o : \mathcal{D}_{*} \to D_{*}
  \end{aligned}
\]

\[
  (207) \quad
  \begin{cases}
    P_{d_0,d_1,\dots,d_r} = \text{image inverse de } P_{d_0} \text{ sur } D_{d_0} \\
    \qquad \text{par } D_{d_0,\dots,d_r} \to D_{d_0} \\
    P^{*}_{d_0,d_1,\dots,d_r} = P_{d_0,\dots,d_r} \mid D^{*}_{d_0,\dots,d_r}
  \end{cases}
\]

\[
  (208) \quad
  \begin{cases}
    G_{d_0,\dots,d_r} = \text{image inverse de } G_{d_0} \text{ sur } D_{d_0}
    \text{ par } D_{d_0,\dots,d_r} \to D_{d_0} \\
    G^{*}_{d_0,\dots,d_r} = G_{d_0,\dots,d_r} \mid D^{*}_{d_0,\dots,d_r} .
  \end{cases}
\]

\[
  (210) \quad
  \begin{cases}
    T_{d_0,\dots,d_r} = \text{image inverse de } T_{d_0} \text{ (sur } D_{d_0}\text{)}
    \text{ par } D_{d_0,\dots,d_r} \to D_{d_0} \\
    \qquad \simeq \text{syst. local des } \pi_1 \text{ (ou des } H_1)
    \text{ des fibres de } P_{d_0 \dots d_r} \\
    T^{*}_{d_0 \dots d_r} = T_{d_0 \dots d_r} \mid D^{*}_{d_0,\dots,d_r} ,
  \end{cases}
\]

\[
  \text{\struck{$\Pi_{\Delta_r} = \coprod \Pi_{d_0,\dots,d_r}$}}
\]

\[
  (211) \quad
  \begin{cases}
    G_{\Delta_r} = \coprod\limits_{\substack{(d_0,\dots,d_r) \in \mathbb{N}^{r+1} \\ d_0 \geqslant d_1 \geqslant \dots \geqslant d_r}} G_{d_0,\dots,d_r}
    & \text{groupe sur } D_{\Delta_r} \\
    P_{\Delta_r} = \coprod P_{d_0,\dots,d_r} & \text{torseur sous } G_{\Delta_r} \\
    T_{\Delta_r} = \coprod T_{d_0,\dots,d_r} & \text{système local des } \pi_1 \text{ ou } H_1
    \text{ de } G_{\Delta_r} \text{ ou } P_{\Delta_r} \text{ au choix}
  \end{cases}
\]

\[
  (212) \quad
  \begin{cases}
    \Pi_{\Delta_r} = \Pi_1 D^{*}_{\Delta_r} & \text{groupoïde fond. de } D^{*}_{\Delta_r} \\
    \widetilde{\Pi}_{\Delta_r} = \Pi_1 P^{*}_{\Delta_r} & \text{id.\ de } P^{*}_{\Delta_r}
  \end{cases}
\]

\[
  (213) \qquad \widetilde{\Pi}_{\Delta_r} \longrightarrow \Pi_{\Delta_r}
\]

\[
  (214) \quad
  \begin{cases}
    \Pi_{\Delta_r} = \coprod\limits_{d_0 \geqslant d_1 \geqslant \dots \geqslant d_r} \Pi_{d_0,d_1,\dots,d_r} \\
    \widetilde{\Pi}_{\Delta_r} = \coprod\limits_{d_0 \geqslant \dots \geqslant d_r} \widetilde{\Pi}_{d_0,d_1,\dots,d_r}
  \end{cases}
\]

\[
  (215) \qquad \Delta_r \xrightarrow{\ \alpha\ } \Delta_{r'} \qquad \text{tel que } \alpha(0) = 0
\]

\[
  \widetilde{\Pi}_{d_0 d_1 \dots d_r} \longrightarrow \Pi_{d_0 \dots d_r}
\]

\[
  (218) \qquad \Delta_r \xrightarrow{\ \alpha\ } \Delta_{r'}
\]

\[
  (219) \qquad \boxed{\ \widetilde{\Pi}_{\Delta_{r'}} \xrightarrow{\ \widetilde{\alpha}^{*}\ } \widetilde{\Pi}_{\Delta_r}\ }
\]

\[
  D^{*}_{d'_0,\dots,d'_{r'}} \longrightarrow D^{*}_{d_0,\dots,d_r}
\]

\[
  d_{*} = \{d_0 > d_1 > \dots > d_r\}, \qquad \text{et si}
\]

\[
  (221) \qquad d'_{*} = \{d'_0 > d'_1 > \dots > d'_{r'}\}
\]

\[
  (222) \qquad \Pi_1 P^{*}_{d_0,d_1,\dots,d_r} \longrightarrow \Pi_1 P^{*}_{d'_0,\dots,d'_r}
\]

\[
  (223) \qquad D^{*}_{d_0,\dots,d_r} \longrightarrow D_{d'_0,\dots,d'_{r'}}
\]

\[
  (225) \qquad d = d_0 - d'_0
\]

\[
  (225) \qquad D^{*}_{d_{*}} \longrightarrow D_{d'_{*}}
\]

\[
  (226) \qquad \Theta^{d*}_{d'_*} \overset{\text{déf}}{=} \Theta^{d}_{d'_*} \setminus \Theta^{d+1}_{d'_*}
\]

\[
  (227) \qquad \Theta^{(d)}_{d'_*} \xleftarrow{\ \sim\ } D_{\tilde{d}'_*}
  \qquad \text{où } \tilde{d}'_* = (d_0, d'_0, \dots, d'_r)
\]

\[
  (228) \qquad D^{*}_{\tilde{d}'_*} \xrightarrow{\ \sim\ } \Theta^{(d)*}_{d'_*} ,
\]

\[
  (229) \qquad D_{d_*} \longrightarrow D_{\tilde{d}'_*} \simeq \Theta^{(d)*}_{d'_*} \longrightarrow D_{d'_*}
\]

\[
  (230) \qquad D''^{*} \longrightarrow D' \longrightarrow \text{\struck{$X$}}\ Y
\]

\[
  (231) \qquad \boxed{\ \Pi_1 P_{D''^{*}} \xrightarrow{\ r_{D'',D'}\ } \Pi_1 P_{D'^{*}}\ }
\]

\[
  P_{D'^{*}} = P_{D'} \mid D'^{*} \hookrightarrow P_{D'}
\]

\[
  (232) \qquad \Pi_1 P_{D'^{*}} \longrightarrow \Pi_1 P_{D'}
\]

\[
  (233) \qquad \Pi_1 P_{D'^{*}} \longrightarrow \Pi_1 P_{D'} ,
\]

\[
  P_{D'} \longrightarrow P_{D''} .
\]

\[
(2.35)\qquad
\begin{array}{ccccc}
D'' & \longrightarrow & D' & & \\
\downarrow & & \downarrow & & \\
V_{D'',Y} & \xrightarrow[\text{imm. loc.}]{} & V_{D',Y} & \xrightarrow[\text{imm. loc.}]{} & Y
\end{array}
\]

\[
(2.36)\qquad
\begin{array}{ccccc}
V^*_{D'',Y} & \longrightarrow & V^*_{D',Y} & \longrightarrow & Y\\
\| & & \| & & \\
V_{D'',Y}\setminus\Theta_{D'',Y} & & V_{D',Y}\setminus\Theta_{D',Y} & &
\end{array}
\]

\[
(2.37)\qquad \Pi_1 V^*_{D'',Y}\longrightarrow \Pi_1 V^*_{D',Y}
\]

\[
(2.38)\qquad D'''\to D''\to D'\to Y .
\]

\[
(2.39)\qquad \Pi_1 V^*_{D',Y}\xleftarrow{\ \approx\ }\Pi_1 P_{D'^*}
\]

\[
(2.40)\qquad V_{D',Y}\hookrightarrow \check N_{D',Y}
\]

\[
\text{\struck{$\Pi_1 V^*_{D',Y}\longrightarrow \Pi_1 P_{D'^*}$}}
\]

\[
(2.41)\qquad V^*_{D',Y}\longrightarrow P_{D'^*}
\]

\[
(2.42)\qquad \Pi_1 V^*_{D',Y}\longrightarrow \Pi_1 P_{D'^*} .
\]

\[
(2.43)\qquad \Pi_1^{\text{\uncertain{arcs}}} P_{D'^*}\longrightarrow \Pi_1^{\text{rev}} V^*_{D',Y}\ )
\]

\[
(2.44)\qquad \Theta^{(0)}=X\supset\Theta^{(1)}=\Theta\supset\Theta^{(2)}\supset\Theta^{(3)}\supset\cdots
\]

\[
D_d=\widetilde{\Theta^{(d)}}\quad\text{« normalisé topologique » de }\Theta^{(d)},
\]

\[
\text{\struck{$D^*_d=\Theta^{(d)}\setminus\Theta^{(d\pm1)}$}}
\]

\[
(2.45)\qquad \text{\struck{$V_{D^*_d}$ et $V^*_{D^*_d}\hookrightarrow V_{D^*_d}$}}
\]

\[
(2.45)\qquad
\begin{array}{ccl}
 & & V^*_{d_0\cdots d_r}\overset{\text{déf}}{=}V^*_{D_{d_0\cdots d_r},X}\qquad\text{où }X=D_0\\
 & & \ \big\downarrow\\
D_{d_0,\dots,d_r} & \to & V_{d_0\cdots d_r}=V_{D_{d_0\cdots d_r},X}
\end{array}
\]

\[
V^*_{D_{d_0,\dots,d_r,\delta_0,\delta_1,\dots,\delta_s}}\qquad(\text{pour }d_0>\cdots>d_r>\delta_0
\]

\[
\text{\struck{$V^*_{\Theta,X}\hookrightarrow X^*$, \quad $V^*_{\Theta,X}\overset{\text{déf}}{=}X^{*\infty}$, \quad $X^*=X\setminus\Theta$}}
\]

\[
\text{\struck{$\varinjlim\big(\widetilde\Pi_{d_0}\ (1\le d_0\le3),\ \widetilde\Pi_{d_0d_1}\ (d_0>d_1\geq$}}
\]

\[
(2.46)\qquad \Pi_1\,\text{\struck{$X$}}^{*\infty}\simeq\varinjlim\big(\widetilde\Pi_1\leftarrow\widetilde\Pi_{2,1}\rightarrow\widetilde\Pi_2\big)
\]

\[
(2.47)\qquad \Pi_1 X\simeq\varinjlim\big(\widetilde\Pi_1\leftarrow\widetilde\Pi_{2,1}\rightarrow\widetilde\Pi_2\big).
\]

\[
\text{\struck{$D_i^{*\infty}\hookrightarrow$ \ill{}, \quad $D_i^{*\infty}\overset{\text{déf}}{=}V^*_{\Theta_i,D_i}\setminus\Theta_i$}}
\]

\[
(2.48)\qquad D_d^\Sigma\subset D_d
\]

\[
D_{d,d'}^\Sigma\supset\text{l'image inverse de }D_{d'}^\Sigma,\qquad D_{d,d'}^\Sigma=\text{l'image inverse de }D_d^\Sigma
\]

\[
(2.50)\qquad D_{d,d'}^\Sigma=\text{image inverse de }D_{d'}^\Sigma\text{ par }D_{d,d'}\to D_{d'}
\]

\[
(2.52)\qquad \pi_0(D_d^\Sigma)\overset{\text{déf}}{=}\pi_0(D_d)^\Sigma\subset\pi_0(D_d),
\]

\[
(2.53)\qquad D_{d_0\cdots d_r}^\Sigma=\text{image inverse de }D_{d_r}^\Sigma\text{ dans }D_{d_0,\dots,d_r}
\]

\[
(2.54)\qquad \widetilde\Pi_0\longleftarrow\widetilde\Pi_{1,0}\longrightarrow\text{\struck{$\Pi_{1,0}$ \ill{}}}
\]

\[
(2.55)\qquad \Sigma\subset\Sigma'\ \text{\struck{\ill{}}}
\]

\[
(2.56)\qquad \text{\struck{\ill{}}}\ \Sigma'\subset\Sigma''\ \text{et un}\ \Sigma_1\subset\Sigma''
\]

\[
V_{\Sigma',\Sigma''}\cap\Sigma_1=V_{\Sigma'\cap\Sigma_1,\Sigma_1})
\]

\[
(2.57)\qquad V^{\Sigma_1}_{\Sigma',\Sigma''}\overset{\text{déf}}{=}V_{\Sigma',\Sigma''}\setminus V_{\Sigma',\Sigma''}\cap\Sigma
\]

\[
\Sigma'_0\subset\Sigma'
\]

\[
(2.58)\qquad V^{\Sigma\setminus\Sigma\cap\Sigma'_0}_{\Sigma'\setminus\Sigma'_0,\ \Sigma''\setminus\Sigma'_0},
\]

\[
V^{\Sigma\cup\Sigma_0}_{\Sigma',\Sigma''}
\]

\[
V^{\Sigma_0}_{\Sigma',X}\quad\text{et}\quad V_{\Sigma'\setminus\Sigma_0,\ X\setminus\Sigma_0}
\]

\[
(2.59)\qquad V_{\Sigma',\Sigma''}\longrightarrow V_{\Sigma'_1,\Sigma''_1}\quad\text{pour}\quad(\Sigma',\Sigma'')\hookrightarrow(\Sigma'_1,\Sigma''_1),
\]

\[
\begin{array}{c}(X,\Theta)\\ \|\ \ \|\\ D_0\ \Theta_0\end{array}\quad\text{par un}\quad(D_{d_*},\Theta_{d_*})\ \text{quelconque}
\]

\[
d'_*=\{d'_0>\cdots>d'_r\geq0\}\hookrightarrow d_*=\{d_0>\cdots>d_r\geq0\}
\]

\[
(2.60)\qquad V_{d'_*,d'_*}\overset{\text{déf}}{=}V_{D^*_{d^*_*},\,D_{d'_*}}
\]

\[
(2.61)\qquad V_{d_*,d_*}=D^*_{d_*}=D^*_{d_0,\dots,d_r},
\]

\[
V_{(d_1,\dots,d_{r-1},0),\,\Theta}=V_{(d_1,\dots,d_{r-1},0)}=V_{(d_1,\dots,d_{r-1})}
\]

\[ D_{d_*}\longrightarrow D_0=X \]

\[ (261)\qquad X=\mathbb{E}^{1\,\mathrm{an}},\qquad \Theta=\{0\} \]

\[ (262)\quad \left\{\begin{array}{l} D_0=\mathbb{C},\qquad \boxed{D_0^*=\mathbb{C}^*}\\[2pt] D_{1,0}\xrightarrow{\ \sim\ } D_1=\{\mathbb{P}\},\qquad D_{1,0}^*=D_{1,0}\\ \quad\big\downarrow\ \text{inclusion de $\{0\}$ dans $\mathbb{C}$}\\ D_0=\mathbb{C} \end{array}\right. \qquad D_1=\boxed{D_1^*=\{\mathbb{P}\}} \]

\[ (263)\quad \left\{\begin{array}{ll} \boxed{\mathcal{V}_0^*\simeq\mathcal{V}_0\xrightarrow{\ \simeq\ } D_0^*=\mathbb{C}^*} & \text{fibré trivial sur $\mathbb{C}^*$ (fibre ponctuelle)}\\ \boxed{\mathcal{V}_1=\mathbb{D}} & \text{(germe de disques ouverts centrés en 0)}\\ \boxed{\mathcal{V}_1^*=\mathbb{D}^*} & \text{(germe de disques ouverts épointés en leur}\\ & \text{\quad centre 0)} \end{array}\right. \]

\[ \mathcal{V}_{1,0}\simeq\mathcal{V}_1,\qquad \mathcal{V}_{1,0}^*\simeq\mathcal{V}_1^* \]

\[ (264)\quad \left\{\begin{array}{ll} \mathbb{C}^* & \text{(espaces analytiques « globaux »)}\\ \mathbb{P}\ \text{point} & \text{(\quad id\quad)}\\ \mathbb{D} & \text{germe de disques}\\ \mathbb{D}^* & \text{germe de disques épointés} \end{array}\right. \]

\[ (266)\qquad \mathbb{C}\simeq\mathbb{C}^*\amalg_{\mathbb{D}^*}\mathbb{D}\,. \]

\[ (267)\qquad X=\mathbb{C}^I,\qquad \Theta=\operatorname{div}\Big(\prod_{i\in I}z_i\Big) \]

\[ (268)\quad \left\{\begin{array}{l} I=\underbrace{I'}_{\substack{\text{correspond aux facteurs}\\ \text{du type « global »}}}\amalg\underbrace{I''}_{\substack{\text{correspond aux facteurs}\\ \text{du type local}}}\\[18pt] I'=\underbrace{I'_0}_{\substack{\text{correspond aux facteurs } \mathbb{P}\\ \text{(de dim 0)}}}\amalg\underbrace{I'_1}_{\substack{\text{correspond aux facteurs } \mathbb{C}^*\\ \text{(de dim 1)}}}\\[18pt] I''=\underbrace{I''_0}_{\text{facteurs } \mathbb{D}}\amalg\underbrace{I''^*}_{\text{facteurs } \mathbb{D}^* \text{ (épointés)}} \end{array}\right. \]

\[ d'_0=\operatorname{card} I'_0, \]

\[ d_1=\operatorname{card} I'_1 \]

\[ \text{\struck{$D^*_{I'_1}$}}=\underbrace{\prod_{i\in I'_1}\mathbb{C}^*}_{\mathbb{C}^{*I'_1}}\times\underbrace{\prod_{i\in I\setminus I'_1}\mathbb{P}}_{\mathbb{P}^{I'_0}} \]

\[ (269)\qquad \begin{array}{l} D_J=\mathbb{P}^J\times\mathbb{C}^{I\setminus J}\subset\mathbb{C}^I=X\\ \quad\cup\\ D^*_J=\mathbb{P}^J\times\mathbb{C}^{*\,I\setminus J} \end{array} \]

\[ (270)\qquad \left\{\begin{array}{l} D_d=\coprod_{J\subset\mathcal{P}(I),\ \operatorname{card}J=d} D_J\\ D^*_d=\coprod D^*_J \end{array}\right. \]

\[ (271)\qquad \underset{(\text{« âme »})}{A}=D^*_{\underbrace{\scriptstyle I\setminus I'_1}_{I'_0\amalg I''}}\ \underset{\substack{\text{immersion}\\ \text{de codim. } I''}}{\hookrightarrow}\ D^*_{I'_0}=\underset{(\text{« réceptacle »})}{R} \]

\[ (272)\qquad \begin{array}{ccccc} A & \hookrightarrow & B & \hookrightarrow & R\\ \| & & \| & & \|\\ D^*_{I'_0\amalg I''} & \big| & D^*_{I'_0\amalg I''^*} & \big| & D^*_{I'_0}\\ & \scriptstyle\text{codim. card } I''_0=d''_0 & & \scriptstyle\text{codim.}=\operatorname{card}I''^*=d''^* & \end{array} \]

\[ (273)\qquad \begin{array}{ccc} \mathcal{V}_{A,R} & \overset{\text{can}}{\hookrightarrow} & \mathcal{V}_{B,R}\\ \big\uparrow & & \big\uparrow\\ \mathcal{V}^{B(*)}_{A,R} & \hookrightarrow & \mathcal{V}^*_{B,R} \end{array} \]

\[ (274)\qquad \begin{array}{ccccc} d''_* & \subset & d'_* & \subset & d_*\\ \| & & \| & & \|\\ \{d''_0>\dots>d''_{r''}\} & & \{d'_0>\dots>d'_{r'}\} & & \{d_0>\dots>d_r\} \end{array} \]

\[ (275)\qquad D_{d_*}\longrightarrow D_{d'_*}\longrightarrow D_{d''_*} \]

\[ (277)\quad \left\{\begin{array}{ll} D_0^*=\mathbb{C}^*\times\mathbb{C}^* & (D_0=X=\mathbb{C}\times\mathbb{C})\\[2pt] \left.\begin{array}{l}\Delta_1^*=\mathbb{C}^*\times\mathbb{P}\\ \Delta_2^*=\mathbb{P}\times\mathbb{C}^*\end{array}\right\} & D_1^*=\Delta_1^*\amalg\Delta_2^*\quad (D_1=\Delta_1\amalg\Delta_2)\\ & \quad\text{où } \Delta_1=\mathbb{C}\times\mathbb{P},\ \Delta_2=\mathbb{P}\times\mathbb{C}\\[2pt] \mathbb{P}=\mathbb{P}\times\mathbb{P} & D_2^*=D_2=\mathbb{P}\\[6pt] \mathcal{V}_{\Delta_1^*,X}=\mathbb{C}^*\times\mathbb{D} & \mathcal{V}^*_{\Delta_1^*,X}=\mathbb{C}^*\times\mathbb{D}^*\\ \mathcal{V}_{\Delta_2^*,X}=\mathbb{D}\times\mathbb{C}^* & \mathcal{V}^*_{\Delta_2^*,X}=\mathbb{D}^*\times\mathbb{C}^*\\[2pt] \mathcal{V}_{\mathbb{P},\Delta_1}=\mathbb{D}\times\mathbb{P} & \mathcal{V}^*_{\mathbb{P},\Delta_1}=\mathbb{D}^*\times\mathbb{P}\\ \mathcal{V}_{\mathbb{P},\Delta_2}=\mathbb{P}\times\mathbb{D} & \mathcal{V}^*_{\mathbb{P},\Delta_2}=\mathbb{P}\times\mathbb{D}^*\\[6pt] \mathcal{V}_{\mathbb{P},X}=\mathbb{D}\times\mathbb{D}, & \\ \mathcal{V}^{\Delta_1}_{\mathbb{P},X}=\mathbb{D}\times\mathbb{D}^*, & \mathcal{V}^{\Delta_2}_{\mathbb{P},X}=\mathbb{D}^*\times\mathbb{D}\\ \mathcal{V}^*_{\mathbb{P},X}=\mathbb{D}^*\times\mathbb{D}^* & \end{array}\right. \]

\[ (278)\qquad X=\mathbb{C}^2,\quad \underbrace{\mathbb{C}^2\setminus\Delta_1}_{U_1},\quad \underbrace{\mathbb{C}^2\setminus\Delta_2}_{U_2},\quad U=U_1\cup U_2=\mathbb{C}^2-\{\mathbb{P}\},\quad U_1\cap U_2=\mathbb{C}^*\times\mathbb{C}^* \]

\[ \left.\begin{array}{l} U_1=\mathbb{C}\times\mathbb{C}^*\\ U_2=\mathbb{C}^*\times\mathbb{C}\end{array}\right\}\quad \mathbb{C}=\mathbb{C}^*\amalg_{\mathbb{D}^*}\mathbb{D} \]

\[ U=U_1\amalg_{U_1\cap U_2}U_2\qquad\text{où}\quad U_1\cap U_2=D_0^* \]

\[ X=\mathbb{C}\times\mathbb{C} \]

\[ D_{d_*}\longrightarrow D_0 \]

\[ (282)\qquad \begin{array}{ccccccccccc} \check{N}_{d_*} & \overset{\text{déf}}{=} & \check{N}_{d_*,d_0} & \supset & \check{N}_{d_*,d_1} & \supset & \check{N}_{d_*,d_2} & \supset\cdots\supset & \check{N}_{d_*,d_r} & \supset & \check{N}_{d_*,0}\\ & & \|\,{\scriptstyle\text{déf}} & & & & & & & & \|\\ & & \check{N}_{D_{d_*},D_{d_0}} & & & & & & & & \{0\}_{D_{d_*}}\\ & & \text{rg } d_0 & & \text{rg } d_1 & & \text{rg } d_2 & & \text{rg } d_r & & \end{array} \]

\[ (283)\qquad \check{N}_{d_*,d_i}=\check{N}_{D_{d_*},D_{d_i}}, \]

\[ (284)\qquad 0\longrightarrow\check{N}_{d_*,d_i}\longrightarrow\check{N}_{d_*}\longrightarrow(\check{N}_{d_i}\,|\,D_{d_*})\longrightarrow 0\,; \]

\[ (285)\qquad 0\longrightarrow\underset{\substack{\wr\\ \check{N}_{d_*,d'_0}}}{\check{N}_{D_{d_*},D_{d'_*}}}\longrightarrow\underset{\substack{\|\,\text{déf}\\ \check{N}_{d_*}}}{\check{N}_{D_{d_*},D_0}}\longrightarrow(\check{N}_{D_{d'_*}}\,|\,D_{d_*})\longrightarrow 0 \]

\[ (286)\qquad \check{N}_{d_*;(d'_0,d''_0)}\overset{\text{déf}}{=}\check{N}_{d_*,d''_0}\big/\check{N}_{d_*,d'_0} \]

\[ d'_0,\ d''_0\in d_*,\qquad d''_0\leqslant d'_0\ (\leqslant d_0), \]

\[ (287)\qquad 0\longrightarrow\check{N}_{d_*,d'_0}\longrightarrow\check{N}_{d_*,d''_0}\longrightarrow\check{N}_{d_*;(d'_0,d''_0)}\longrightarrow 0\,. \]

\[ \begin{array}{ccccc} d''_* & \subset & d'_* & \subset & d_*\\ \| & & \| & & \\ \{d''_0>\dots>d''_{r''}\} & & \{d'_0>\dots>d'_{r'}\} & & \end{array} \]

\[ (288)\qquad \begin{array}{ccccccccc} 0 & \to & \check{N}_{D_{d_*},D_{d'_*}} & \to & \check{N}_{D_{d_*},D_{d''_*}} & \to & (\check{N}_{D_{d'_*},D_{d''_*}}\,|\,D_{d_*}) & \to & 0\\ & & \wr & & \wr & & \wr & & \\ & & \check{N}_{d_*,d'_0} & & \check{N}_{d_*,d''_0} & & \check{N}_{d_*;(d'_0,d''_0)} & & \end{array} \]

\[ (289)\qquad N^*\subset N,\qquad\text{en particulier}\quad \left\{\begin{array}{l} N^*_{d_*}\subset N_{d_*}\\ N^*_{d_*,d_i}\subset N_{d_*,d_i}\\ N^*_{d_*;(d'_0,d''_0)}\subset N_{d_*;(d'_0,d''_0)} \end{array}\right. \]

\[ (290)\qquad \check{N}^{d'_0}_{d_*,d''_0}=\text{image inverse de }\check{N}^*_{d_*;(d'_0,d''_0)}\text{ dans }\check{N}_{d_*,d''_0}, \]

\[ \check{N}_{d_*,d''_0}\longrightarrow\check{N}_{d_*;(d'_0,d''_0)}\quad\text{de (287)}. \]

\[ (291)\qquad \check{N}^{d'_0}_{d_*,d''_0}\ \text{est fibré sur}\ \check{N}^*=\check{N}^*_{d_*;(d'_0,d''_0)}, \]

\[ (292)\qquad \check{N}^{d'_0}_{d_*,d''_0}\,|\,D^*_{d_*}\ \simeq\ \mathcal{V}^{d'_0}_{d_*,d''_0} \]

\[ (293)\qquad \check{N}^{d'_0}_{d_*,d''_0}\,|\,D^*_{d_*}\longrightarrow\check{N}^*_{d_*;(d'_0,d''_0)}\,|\,D^*_{d_*} \]

\[ (294)\qquad \boxed{\ \mathcal{V}^{d'_0}_{d_*,d''_0}\ \simeq\ \check{N}^*_{d_*;d'_0,d''_0}\,|\,D^*_{d_*}\quad\text{équivalence d'homotopie}\ } \]

\[ (295)\qquad \boxed{\ \Pi_1\big(\underbrace{\check{N}^*_{d_*;(d'_0,d''_0)}\,|\,D^*_{d_*}}_{\overset{\text{déf}}{=}\ \check{N}^{**}_{d_*;(d'_0,d''_0)}}\big)\xrightarrow{\ \approx\ }\Pi_1\mathcal{V}^{d'_0}_{d_*,d''_0}\ } \]

\[
  U \longmapsto \mathrm{Strglob}(U)
\]

\[
  E \longrightarrow X \tag{296}
\]

\[
  D^{*}_{\Delta_0} \longrightarrow X , \tag{297}
\]

\[
  D^{*}_{\Delta_0} \simeq \coprod_{d \in \mathbb{N}} D^{*}_{d} \tag{298}
\]

\[
  \Sigma_* = \{\overline{X_{i_0}} \subset \overline{X_{i_1}} \subset \dots \subset \overline{X_{i_r}}\} \tag{299}
\]

\[
  x \in X_{i_0} \tag{300}
\]

\[
  D^{*}_{\Delta_r} \longrightarrow X \tag{301}
\]

\[
  D^{*}_{\Delta_r} = \coprod_{d_r \in \mathcal{P}_{r+1}(\mathbb{N})} D^{*}_{d_r}
\]

\[
  D^{*}_{\Delta_r} \longrightarrow D_{\Delta_{r'}} \quad \text{pour } \Delta_{r'} \to \Delta_r
\]

\[
  x \in \overline{X_{i_0}} .
\]

\[
  D_{\Delta_r} \longrightarrow X , \tag{302}
\]

\[
  D_{\Delta_2} \xrightarrow{\ \sim\ } D_{\Delta_1} \times_{D_{\Delta_0}} D_{\Delta_1} \tag{303}
\]

\[
  \Sigma_* = \{x \in \overline{X_{i_0}} \subset \overline{X_{i_1}} \subset \overline{X_{i_2}}\}
\]

\[
  \left\{
  \begin{aligned}
    &\text{espace des objets} = D_{\Delta_0} \\
    &\qquad = \text{espace des strates locales pas strictes} \\
    &\text{espace des flèches} = D_{\Delta_1} \\
    &\qquad = \text{espace des drapeaux } x \in \overline{X_{i_0}} \subset \overline{X_{i_1}}
  \end{aligned}
  \right. \tag{303}
\]

\[
  D_{\Delta_r} \xrightarrow{\ \sigma\ } D_{\Delta_0} \tag{304}
\]