\[
  H^0(\hat{X}', \mathcal{O}_{\hat{X}'}) \simeq H^0(X', \mathcal{O}_{X'}) \quad (\ill{}) \ \ill{}
\]

\[
  Z' \cap V(f_{l+1}) \cap \cdots \cap V(f_k) = \emptyset,
\]

\[
  Y' = V(f_1) \cap \cdots \cap V(f_k) \cap X'
\]

\[
  \dim \mathcal{O}_{X,x} = \underbrace{\dim \mathcal{O}_{Y,y_0}}_{n} + \underbrace{\dim X_1}_{\geq k+1}
\]

\[
  (*) \qquad H^i(X, F) \longrightarrow H^i(X_{/Y}, F_{/Y})
\]

\[
  H^0(X, F) \xrightarrow{\ \sim\ } H^0(X_{/Y}, F_{/Y})
\]

\[
  \varinjlim_{U \supset Y} \mathcal{L}(U) \longrightarrow \mathcal{L}(X_{/Y})
\]

\[
  H^1(X, F) \longrightarrow H^1(X_{/Y}, F_{/Y})
\]

\[
  H^i(X, L_\alpha) \simeq H^i(X_{/Y}, L_{\alpha/Y}) \quad \text{pour } i \leq n, \qquad \text{inj pour } i = n+1.
\]

\[
  0 \to F \to L \to G \to 0
\]

\[
  H^0(U, F) \longrightarrow H^0(U_{/Y}, F_{/Y})
\]

\[
  x \in U, \quad \overline{x}^{(U)} \cap Y = \emptyset \Longrightarrow x \notin \operatorname{Ass} \mathcal{O}_X ,
\]

\[
  (\mathrm{LG})_0 \Longleftrightarrow (\mathrm{LG})_{\text{faible}} \Longleftrightarrow (\mathrm{LG})_X \Longrightarrow H^0(X, \mathcal{O}_X(n)) \simeq H^0(X, \mathcal{O}_X(n)_{/Y})
\]

\[
  H^0(X, \mathcal{O}_X) \longrightarrow H^0(X_{/Y}, \mathcal{O}_{X/Y})
\]

\[
  \pi_1(Y) \longrightarrow \varprojlim_{U \text{ voisinage ouvert de } Y} \pi_1(U)
\]

\[
  U' \times_U Y \simeq Y' .
\]

\[
  \varinjlim_{U \supset Y} \operatorname{Pic}(U) \longrightarrow \operatorname{Pic}(X_{/Y})
\]

\[
  \varinjlim_{U \ni Y} \operatorname{Pic}(U) \xrightarrow{\ \sim\ } \operatorname{Pic}(Y) .
\]

\[
  Y' = f^{-1}(Y) ,
\]

\[
  (*) \qquad H^i(X', F') \longrightarrow H^i(X'_{/Y'}, F'_{/Y'})
\]

\[
  F'_{/Y'} \simeq \mathcal{F}' .
\]

\[
  U'_{/Y'} \simeq \mathfrak{X}' \quad (\text{où } Y' = f^{-1}(Y)).
\]

\[
  \hat{X} = X_{/Y}, \qquad \hat{X}' = X'_{/Y'},
\]

\[
  F \mapsto \hat{F} : \mathcal{C}(X') \longrightarrow \mathcal{C}(\hat{X}') .
\]

\[
  \rho : \mathcal{C}(X') / \mathcal{C}_\Phi(X') \longrightarrow \mathcal{C}(\hat{X}') .]
\]

\[
  \operatorname{Hom}(F, G) \simeq \operatorname{Hom}(\hat{F}, \hat{G}) .
\]

\[
  \Gamma F \simeq \Gamma \hat{F}
\]

\[
  0 \to P' \to F' \to \bar{F}' \to Q' \to 0
\]

\[
  0 \to K' \to F' \to G' \to 0
\]

\[
\begin{array}{ccccccc}
  0 \to \Gamma K' & \longrightarrow & \Gamma F' & \longrightarrow & \Gamma G' & \longrightarrow & H^1 K' \\
  \downarrow \wr & & \downarrow & & \downarrow \wr\,? & & \wr \\
  0 \to \Gamma \hat{K}' & \longrightarrow & \Gamma \hat{F}' & \longrightarrow & \Gamma \hat{G}' & \longrightarrow & H^1 \hat{K}'
\end{array}
\]

\[
  H^i(U, F) \longrightarrow H^i(\hat{X}, \hat{F})
\]

\[
  \begin{array}{l}
  H^i_Z(X, F(-m)) \xrightarrow{\alpha_i} H^i(X, F(-m)) \to H^i(U, F(-m)) \\
  \qquad \to H^{i+1}_Z(X, F(-m)) \xrightarrow{\alpha_{i+1}} H^{i+1}(X, F(-m))
  \end{array}
\]

\[
  \prod_{z \in Z} E^{r-i}_z(m)^{\wedge} \longleftarrow \Gamma E^{r-i}(m)
  \qquad\qquad
  \prod_{z \in Z} E^{r-i-1}(m)_z^{\wedge} \longleftarrow \Gamma E^{r-i-1}(m)
\]

\[
\left.
\begin{array}{l}
  \alpha_i \text{ injectif pour } m \text{ grand} \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ isolé dans } (\operatorname{Supp} E^{r-i} \cup \lbrace z \rbrace) \\
  \Updownarrow \\
  H^{i-1}(X, F(-m)) \to H^{i-1}(U, F(-m)) \text{ surjectif pour } m \text{ grand}
\end{array}
\right.
\]

\[
(*) \quad
\left.
\begin{array}{l}
  \alpha_i \text{ injectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \exists \text{ voisinage ouvert } V \text{ de } Z, \text{ tel que } F \text{ soit de profondeur } \ldots \text{ sur } U \cap V \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand},\ i < n
\end{array}
\right.
\]

\[
  \coprod_{\substack{p \geq 0 \\ q \geq 0}} F_0(q - p)
  \qquad\qquad
  \mathcal{M} + \mathcal{M}(1) + \cdots + \mathcal{M}(-p)
\]

\[
  \coprod_{q} S_q
\]

\[
  H^i(X, F) \longrightarrow H^i(\hat{X}, \hat{F})
\]

\[
\left.
\begin{array}{l}
  \alpha_i \text{ bijectif pour } m \text{ grand} \\
  \Updownarrow \\
  \operatorname{Supp} E^{r-i} \subset Z \\
  \Updownarrow \\
  H^i_x(F_x) = 0 \text{ pour tt } x \text{ fermé dans } U.
\end{array}
\right.
\]

\[
\left.
\begin{array}{l}
  \alpha_i \text{ bijectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \operatorname{Supp} E^{r-i} \subset Z \text{ pour } i \leq n \\
  \Updownarrow \\
  \operatorname{prof}(F_x) > n \text{ pour tt } x \text{ fermé dans } U \\
  \Updownarrow \\
  H^i(U, F(-m)) \leftarrow H^i(X, F(-m)) \text{ pour } m \text{ grand},\ i < n \text{ et} \\
  H^n(U, F(-m)) \to H^{n+1}_Z(X, F(-m)) \text{ injectif}
\end{array}
\right.
\]

\[
(\mathrm{B}_n) \quad
\left.
\begin{array}{l}
  \alpha_i \text{ bijectif pour } m \text{ grand},\ i \leq n, \\
  \quad \text{et } \alpha_i \text{ injectif pour } m \text{ grand},\ i = n+1 \\
  \Updownarrow \\
  \operatorname{Supp} E^{r-i} \subset Z \text{ pour } i \leq n,\ \operatorname{Supp} E^{r-n-1} \ldots Z \\
  \Downarrow \\
  \operatorname{prof}(F_x) > n \text{ si } x \text{ fermé dans } U, \text{ et} \\
  \operatorname{prof}(F_x) > n+1 \text{ si de plus } x \in V \cap U,\ V \text{ voisinage ouvert convenable de } Z \\
  \Updownarrow \\
  H^i(U, F(-m)) = 0 \text{ pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S \text{ pour } i \leq n
\end{array}
\right.
\]

\[
\left\lbrace
\begin{array}{l}
  H^i(U, F) \simeq H^i(\hat{X}, \hat{F}) \quad \text{est} \quad
  \left\lbrace
  \begin{array}{l}
    \text{isom si } i < n \\
    \text{mono si } i = n
  \end{array}
  \right. \\[1ex]
  H^i(\hat{X}, \hat{F}) \simeq \varprojlim H^i(X_m, F_m) \text{ pour } i \leq n
\end{array}
\right.
\]

\[
  H^0(U, F) \longrightarrow H^0(\hat{X}, \hat{F})
\]

\[
  H^i(U, F) \longrightarrow H^{i+1}_Z(X, F)
\]

\[
(\mathrm{A}'_i) \quad
\left.
\begin{array}{l}
  H^i(U, F) \text{ de dim finie sur } k \\
  \Updownarrow \\
  H^{i+1}_z(F_z) \text{ de dim finie [i.e.\ de long.\ finie] pour tt } z \in Z \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ pt isolé de } \operatorname{Supp} E^{r-i-1} \cup \lbrace z \rbrace \\
  \Updownarrow \\
  \alpha_{i+1} \text{ injectif pour } m \text{ grand} \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand}
\end{array}
\right.
\]

\[
\left.
\begin{array}{l}
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S \text{ pour } i \leq n \\
  \Updownarrow \\
  R^i g_*(F) \text{ coh.\ pour } i \leq n, \text{ i.e.\ } \underline{H}^i_Z(F) \text{ coh.\ pour } i \leq n+1, \\
  \quad \text{i.e.\ } H^i_z(F_z) \text{ de dim finie pour } i \leq n+1,\ z \in Z
\end{array}
\right.
\]

\[
(\mathrm{A}^f_n) \quad
\left.
\begin{array}{l}
  H^i(U, F) \text{ de dim finie si } i \leq n \\
  \Updownarrow \\
  F \text{ de prof} > n+1 \text{ sur } U \text{ au voisinage de } Z \\
  \Updownarrow \\
  z \in Z \Rightarrow z \text{ pt isolé de } \operatorname{Supp} E^{r-i-1} \cup \lbrace z \rbrace \text{ si } i \leq n \\
  \Updownarrow \\
  \alpha_i \text{ injectif pour } \underline{m \text{ grand}} \text{ si } i \leq n+1 \\
  \Updownarrow \\
  H^i(X, F(-m)) \to H^i(U, F(-m)) \text{ surjectif pour } m \text{ grand},\ i \leq n \\
  \Updownarrow \\
  R^i g_*(F) \text{ coh.\ pour } i \leq n \iff \underline{H}^i_z(F) \text{ de dim finie pour } z \in Z,\ i \leq n+1 \\
  \Updownarrow \\
  \coprod_{m \geq 0} H^i(U, F(m)) \text{ de type fini sur } S \text{ pour } i \leq n
\end{array}
\right.
\]

\[
  \sum_{p \geq 0} R^i f_{0*} F_0(q - p) \qquad\qquad q - p \geq n
\]

\[
  S_{pq} = \left\lbrace
  \begin{array}{ll}
    0 & \text{si } p > 0 \\
    S_q & \text{si } p = 0
  \end{array}
  \right.
\]

\[
  0 \to H^0_a(X, \mathcal{O}_X^*) \to H^0(X, \mathcal{O}_X^*) \to H^0(U, \mathcal{O}_X^*) \to H^1_a(X, \mathcal{O}_X^*) \to H^1(X, \mathcal{O}_X^*) \to \cdots
\]

\[
  \cdots \to H^1(U, \mathcal{O}_X^*) \to H^2_a(X, \underline{\mathcal{O}}_X^*) \to H^2(X, \underline{\mathcal{O}}_X^*)
\]

\[
  1 - g = \sum (1 - g_i) - \sum n_s , \qquad 1 - g = \nu - \sum g_i - \sum n_s , \qquad g = 1 + \sum g_i + \sum n_s - \nu
\]

\[
  1 - g = 1 - g' - \sum n_s , \qquad g = g' + \sum n_s
\]

\[
  1 - g = 2 - g' - g'' - \sum n_s , \qquad g = (g' + g'') + \sum n_s - 1
\]

\[
  \boxed{\operatorname{Pic}(\hat{C}) \simeq \mathbf{Z} \xrightarrow{\ \sim\ } \operatorname{Ker}\bigl[\operatorname{Pic}(X') \xrightarrow{i^*} \operatorname{Pic}(X)\bigr]}
\]

\[
  \operatorname{Pic}(X) \xrightarrow{\ \sim\ } \operatorname{Pic}(C_X) \quad (?)
\]