\[ f \colon X \longrightarrow S \]

\[ U = X - X_0 . \]

\begin{equation*}
\tag{1} D \longrightarrow \mathrm{Pic}(X) \longrightarrow \mathrm{Pic}(U) \longrightarrow 0 ,
\end{equation*}

\begin{equation*}
\tag{2} \mathrm{Pic}(X) \simeq \varprojlim_n \mathrm{Pic}(X_n) ,
\end{equation*}

\begin{equation*}
\tag{3} \mathrm{Pic}(X_n) \xrightarrow{\ \sim\ } P_n(k) ,
\end{equation*}

\begin{equation*}
\tag{4} \mathrm{Pic}(X) \xrightarrow{\ \sim\ } P(k) .
\end{equation*}

\begin{equation*}
\tag{5} D_k \longrightarrow P ,
\end{equation*}

\begin{equation*}
\tag{6} \mathrm{Pic}(U) \hookrightarrow Q(k) ,
\end{equation*}

\begin{equation*}
\tag{7} P_n = \underline{\mathrm{Pic}}_{X_n/k} .
\end{equation*}

\begin{equation*}
\tag{8} \mathrm{Pic}(X_n) \hookrightarrow P_n(k) ,
\end{equation*}

\begin{equation*}
\tag{9} \pi = \mathrm{Gal}(\bar{k}/k) ,
\end{equation*}

\begin{equation*}
\tag{10} Q(k) \simeq \mathrm{Pic}(U')^{\pi} ,
\end{equation*}

\begin{equation*}
\tag{11} \underline{\mathrm{Lie}}(P) \simeq H^1(X, \mathcal{O}_X) .
\end{equation*}

\begin{equation*}
\tag{12} D \longrightarrow \mathrm{NS}(X_0)
\end{equation*}

\begin{equation*}
\tag{13} Q = P/\underline{D}
\end{equation*}

\begin{equation*}
\tag{14} P^0 \xrightarrow{\ \sim\ } Q^0 ,
\end{equation*}

\begin{equation*}
\tag{15} Q/Q^0 = (P/P^0)/\underline{D} \simeq \underline{\mathrm{NS}}_{Z_n/k} / \underline{D} ,
\end{equation*}

\begin{equation*}
\tag{16} \underline{\mathrm{Lie}}(Q) \simeq H^1(X, \mathcal{O}_X) .
\end{equation*}

\begin{equation*}
\tag{17} H^1(X, \mathcal{O}_X) \hookrightarrow H^1(U, \mathcal{O}_U) = H^2_s(S)
\end{equation*}

\begin{equation*}
\tag{18} P(T) = \varprojlim P_n(T) \hookleftarrow \varprojlim_n \mathrm{Pic}(X_{n_T}) \text{\add{$/\mathrm{Pic}(T)$}}
\qquad (X_{n\,T} = X_n \times_k T) .
\end{equation*}

\[ C = A \hat{\otimes}_k B = \lim A_n \otimes_k B , \]

\begin{equation*}
\tag{19} \mathrm{Pic}(X \hat{\otimes}_k B) \simeq \varprojlim_n \mathrm{Pic}(X_{nB}) ,
\end{equation*}

\begin{equation*}
\tag{20} B \longmapsto \mathrm{Pic}(X \hat{\otimes}_k B)/\bigl(\text{\add{$\mathrm{Pic}(B) +$}}\, \underline{D}(B)\bigr) \quad (\hookrightarrow Q(B))
\end{equation*}

\begin{equation*}
\tag{21} B \longmapsto \mathrm{Pic}(U \hat{\otimes}_k B)\text{\add{$/\mathrm{Pic}(B)$}} \quad (\hookrightarrow Q(B)) .
\end{equation*}

\begin{equation*}
\tag{22} \mathrm{Pic}(X) \hookrightarrow P(k) ,
\end{equation*}

\begin{equation*}
\tag{23} A \simeq \underline{A}(k) .
\end{equation*}

\[ X_B \overset{\mathrm{dfn}}{=} X \otimes_A \underline{A}(B) , \]

\begin{equation*}
\tag{24} B \longmapsto \mathrm{Pic}(X_B) .
\end{equation*}

\[ f \colon X \longrightarrow S \]

\[ \mathrm{Pic}(X) \longrightarrow \mathrm{Pic}(U) \]

\[
\boxed{\, n^*(L) \simeq L^{\otimes n}\, (\delta L)^{\frac{n(n-1)}{2}} \,}
\qquad\qquad
n + 2\,\frac{n(n-1)}{2} = n^2
\]

\[
\delta L = L^{\otimes 2}\, \delta'(L) \qquad \delta' L_x = L_{2x}\, L_x^{-4}
\]

\[
\boxed{\, n^*(L) \simeq L^{n^2}\, (\delta' L)^{\frac{n(n-1)}{2}} \,}
\]

\[ A \longrightarrow A \times A \qquad A \longrightarrow \hat{A} \]

\[
\boxed{\, \underline{\underline{\mathrm{Pic}}}^{\tau}_{(\prod X_i)/S} \simeq \prod_i \underline{\underline{\mathrm{Pic}}}^{\tau}_{X_i/S} \,}
\]

\[
0 \to \prod_i \underline{\underline{\mathrm{Pic}}}_{X_i/S} \longrightarrow \underline{\underline{\mathrm{Pic}}}_{(\prod X_i)/S} \longrightarrow \prod_{\{i,j\}} \mathcal{C}_S(X_i, X_j) \longrightarrow 0
\]

\[
\begin{array}{ccc}
 & X_i \times X_j & \longrightarrow\ \prod X_i \\
 \nearrow & & \nearrow \\
X_{[ijkl]} \longrightarrow & X_k \times X_l &
\end{array}
\]

\[
X_{ijkl} = \begin{cases}
X_{\emptyset} & \text{si } \{i,j\} \cap \{k,l\} = \emptyset \\
X_{\alpha} & \text{si } \{i,j\} \cap \{k,l\} = \{\alpha\}
\end{cases}
\]

\[ L_{m_1 x_1 + \cdots + m_n x_n} \]

\[ L_{x,y} \simeq L_{xe}\, L_{ye}\, \mathcal{M}_{xy} \qquad \mathcal{M}_{xy}\, \mathcal{M}_{yx} \]

\[
L_{x_1 + \cdots + x_n} = \Bigl(\prod_{i<j} L_{x_i + x_j}\Bigr) \Bigl(\prod L_{x_i}^{-n+2}\Bigr) L_{\emptyset}^{\alpha_n}
\]

\[
L_{x+y} = L_x\, L_y\, (\delta L)_{x,y}
\]

\[
\prod_{i \neq j} L_{x_i + x_j} = \Bigl(\prod L_{x_i}^{n-1}\Bigr) \prod_{i \neq j} (\delta L)_{x_i, x_j}
\]

\[
\boxed{\, L_{x_1 + \cdots + x_n} = \prod_{i<j} (\delta L)_{x_i, x_j} \prod_i L_{x_i}\, L_{\emptyset}^{\alpha_n} \,}
\]

\[
L_{\emptyset}^{-\frac{n(n-1)}{2}}\, L_{\emptyset}^{\frac{(n-1)(n-2)}{2}} = L_{\emptyset}^{-n+1}
\qquad (\delta L)_{0,0} = L_{\emptyset}^{-1}
\]

\[
1 = -\frac{n(n-1)}{2} + n + \frac{(n-1)(n-2)}{2}
\]

\[ 1 + 1 - 1 = 1 \qquad -(n-2) - (n-2) + (n-3) = -(n-1) \]

\[
\alpha_n + \alpha_n - \alpha_{n-1} + 1 = \text{\struck{\ill{}}}\quad 2\alpha_n - \alpha_{n-1} + 1 = \alpha_{n+1}
\]

\[
\alpha_3 = 1 \quad \alpha_4 = 3 \quad \alpha_5 = 6 \quad \alpha_6 = 10 \quad \alpha_7 = 15
\]

\begin{align*}
\alpha_n &= an^2 + bn + c \\
2\alpha_n - \alpha_{n-1} + 1 &= 2an^2 + 2bn + 2c - a(n^2 - 2n + 1) - b(n-1) - c + 1 \\
&= an^2 + (b + 2a)n + (c - a + b + 1) \\
\alpha_{n+1} &= a(n^2 + 2n + 1) + b(n+1) + c \\
&= an^2 + (2a + b)n + (a + b + c)
\end{align*}

\[
a + b + c = -a + b + c + 1 \qquad a = -a + 1 \qquad a = \tfrac12
\]

\[
\alpha_n = \tfrac12 n^2 + bn + c \qquad \alpha_2 = 0,\ \alpha_3 = 1
\]

\[
\tfrac12 \cdot 4 + 2b + c = 0 \qquad \tfrac12 \cdot 9 + 3b + c = 1
\]

\[
2b + c = -2 \qquad 6b + 2c = -7 \qquad 2b = -3 \qquad c = +1
\]

\[
\tfrac12 n^2 - \tfrac32 n + 1 = \tfrac12 [n^2 - 3n + 2]
\qquad
\boxed{\, \alpha_n = \frac{(n-2)(n-1)}{2} \,}
\]

\[
\boxed{\, L_{x_1 \dots x_n} \simeq \Bigl(\prod_{i \neq j} L_{x_i x_j}\Bigr) \Bigl(\prod_i L_{x_i}\Bigr)^{-n+2} L_{\emptyset}^{\frac{(n-1)(n-2)}{2}} \,}
\]

\[
\begin{matrix} x & y & z \\ a & b & c \end{matrix}
\qquad
\boxed{\, L_{x,y,z} \simeq L_{xy}\, L_{y,z}\, L_{x,z}\, L_x^{-1} L_y^{-1} L_z^{-1} L_{\emptyset} \,}
\]

\begin{align*}
L_{xyzt} &= L_{(x,y)z}\, L_{z,t}\, L_{(x,y),t}\, L_{(x,y)}^{-1} L_z^{-1} L_t^{-1} L_{\emptyset} \\
&= L_{xy} L_{y,z} L_{xz} L_x^{-1} L_y^{-1} L_z^{-1} L_{\emptyset}\, L_{zt} \\
&\qquad L_{xy} L_{y,t} L_{x,t} L_x^{-1} L_y^{-1} L_t^{-1} L_{\emptyset}\,
L_{xy}^{-1} L_z^{-1} L_t^{-1} L_{\emptyset}
\end{align*}

\[
\boxed{\, L_{xyzt} = L_{xy} L_{xz} L_{xt} L_{yz} L_{yt} L_{zt}\,
L_x^{-2} L_y^{-2} L_z^{-2} L_t^{-2}\, L_{\emptyset}^{3} \,}
\]

\[
L_{xyztu} = L_{xyzt}\, L_{tu}\, L_{xyzu}\, L_{xyz}^{-1} L_t^{-1} L_u^{-1} L_{\emptyset}
\]

\[
\boxed{\,
\begin{array}{l}
L_{xyztu} = L_{xy} L_{xz} L_{xt} L_{xu} L_{yz} L_{yt} L_{yu} L_{zt} L_{zu} L_{tu} \\
\qquad L_x^{-3} L_y^{-3} L_z^{-3} L_t^{-3} L_u^{-3}\, L_{\emptyset}^{6}
\end{array}\,}
\]

\[
L_{x_1 \dots x_n} = \prod_{i \neq j} L_{x_i x_j} \prod L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n}
\]

\begin{align*}
L_{x_1 \dots x_{n+1}} &= L_{(x_1 \dots x_{n-1}) x_n x_{n+1}} \\
&= L_{x_1 \dots x_n}\, L_{x_1 \dots x_{n-1} x_{n+1}}\, L_{x_n x_{n+1}}\,
L_{x_1 \dots x_{n-1}}^{-1} L_{x_n}^{-1} L_{x_{n+1}}^{-1} L_{\emptyset} \\
&= \prod_{1 \leq i < j \leq n} L_{x_i x_j} \prod_{1 \leq i \leq n} L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n}
\prod_{\substack{1 \leq i < j \leq n+1 \\ i, j \neq n}} L_{x_i x_j}
\prod_{\substack{1 \leq i \leq n+1 \\ i \neq n}} L_{x_i}^{-(n-2)} L_{\emptyset}^{\alpha_n} \\
&\qquad L_{x_n x_{n+1}} \prod_{1 \leq i < j \leq n-1} L_{x_i x_j}^{-1}
\prod_{1 \leq i \leq n-1} L_{x_i}^{+(n-3)} L_{\emptyset}^{-\alpha_{n-1}}\,
L_{x_n}^{-1} L_{x_{n+1}}^{-1} L_{\emptyset}
\end{align*}

\[
\Bigl(\prod_{i \neq j} L_{x_i x_j}\Bigr) \Bigl(\prod L_{x_i}^{-(n-1)}\Bigr) L_{\emptyset}
\]

\[
  f \colon X_1 \times_S X_2 \longrightarrow G
\]

\[
  f = (f_1 p_1)(f_2 p_2)
\]

\[
  f_1 \colon X_1 \to G, \qquad f_2 \colon X_2 \to G
\]

\[
  f = f_1 p_1 + f_2 p_2 + \sigma \varphi
\]

\[
  f_2 = f \circ (\varepsilon_1 \times_S \mathrm{id}_{X_2})
\]

\[
  f' = f - f_2 p_2 \colon X_1 \times_S X_2 \longrightarrow G ;
\]

\[
  f' \colon k(x_1) \otimes_{k(s)} X_2 \longrightarrow G
\]

\[
  f_2 = f \circ (\varepsilon_1 \times \mathrm{id}_{X_2})
\]

\[
  g \mid f^{-1}(U) = \sigma \circ (f \mid f^{-1}(U)) .
\]

\[
  \underline{O}_Y \to h_*(\mathcal{A}) \subset f_*(\underline{O}_X) \simeq \underline{O}_Y ,
\]

\[
  (h \circ g'')_*(\underline{O}_{Z'}) \simeq \underline{O}_Y ,
\]

\[
  \text{\struck{$f_2 \; f = f \circ \varepsilon_1 p_1 - f \circ \varepsilon_2$}} \qquad f' = f - f_1 p_1 - f_2 p_2 \quad (f_i = f \circ u_i \ldots),
\]

\[
  f = u + c
\]

\[
  c = f \circ \varepsilon_A ,
\]

\[
  f = [f \circ (\varepsilon \times \mathrm{id}_A)]\, p_1 + [f \circ (\mathrm{id}_A \times \varepsilon)]\, p_2 + [f \circ (\varepsilon_1 \times \varepsilon_2)]\, p\,,
\]

\[
  X_s \to k(s) \to G .
\]

\[
  X \xrightarrow{\ f\ } S \xrightarrow{\ v\ } G .
\]

\[
  u = u_1\, \mathrm{pr}_1 + u_2\, \mathrm{pr}_2 + v\, p
\]

\[
  u \colon X_1 \times_S X_2 \to G ;
\]

\[
  u = u_1\, \mathrm{pr}_1 + u_2\, \mathrm{pr}_2 .
\]

\[
  x y^{\vee} = e \iff x = y .
\]

\[
  \text{\struck{$f(a,b,c) = [(ab)c]\,[a(bc)]^{\vee}$}}
\]

\[
  f(a,b) = (ab)(ba)^{\vee} .
\]

\[
  f(e, x) = f(x, e) = e ,
\]

\[
  \begin{cases}
    f(a, x) = f(e, x) = e & \text{pour } a \in U \text{ voisinage de } e, \\
    f(x, a) = f(x, e) = e & \text{pour } a \in U',
  \end{cases}
\]

\[
  f(a,b,c) = [(ab)c]\,[a(bc)]^{\vee}
\]

\[
  G_0 = G \times_S S_0
\]

\[
  H^1\bigl(G_0 \times G_0,\ \pi_0^{*}(\underline{\mathcal{V}}_{G_0/S_0}) \otimes \ill{}\bigr) = 
\]

\[
  H^1(G_0 \times G_0, \underline{O}_{G_0 \times G_0}) \otimes_{k} (\mathcal{V} \otimes_k \mathfrak{m}) .
\]

\[
  e\,g = g, \qquad g\,e = g ;
\]

\[
  X_{is'} = \overline{X_{is''}} \cap X_{s'}
\]

\[
  \operatorname{Hom}_{S\text{-gr}}(H, G) \longrightarrow
  \operatorname{Hom}_{S'\text{-gr}}(H', G')
\]

\begin{align*}
  T_X(A) &\simeq \operatorname{Hom}_S(X \times_S X, A)
  \simeq \operatorname{Hom}_{S'\text{-pointés}}(X_{S'}, A_{S'}) \\
  &= \operatorname{Hom}_{S'\text{-pointés}}(X_{S'},
     \operatorname{Pic}^\circ_{B_{S'}/S'}) \\
  &\simeq \mathcal{B}_{S'}(X_{S'}, B_{S'})
  \simeq \operatorname{Hom}_{S'\text{-groupes}}(B_{S'},
     \operatorname{Pic}_{X_{S'}/S'}) \\
  &\simeq \operatorname{Hom}_{S\text{-groupes}}(B,
     \operatorname{Pic}_{X/S})
\end{align*}

\[
  T_X(A) \simeq \operatorname{Hom}_{S\text{-gr}}(\hat{A},
  \operatorname{Pic}_{X/S})
\]

\[
\mathrm{Hom}_{S\text{-gr}}(\hat{B}, \underline{\mathrm{Pic}}_{X/S}) \xleftarrow{\ \sim\ } \mathrm{Hom}_{S\text{-gr}}(B, \underline{B}_{X/S})
\]

\[
\begin{cases}
\mathrm{Hom}_{S\text{-gr}}(\hat{A}, \underline{\mathrm{Pic}}_{X/S}) \xleftarrow{\ \sim\ } \mathrm{Hom}_{S\text{-gr}}(\underline{A}^{0}_{X/S}, A) \\[4pt]
\text{où } \underline{A}^{0}_{X/S} = \hat{\underline{B}}_{X/S}
\end{cases}
\]

\[
\text{\struck{$X$}}\quad P \simeq \underline{A}^{1}_{X/S} \times^{\underline{A}^{0}_{X/S}} (A, u)\ \ldots\ldots
\]

\[
\begin{cases}
X \longrightarrow \underline{A}^{1}_{X/S} \\[4pt]
\underline{\mathrm{Pic}}^{0}_{X/S} \longleftarrow \underline{\mathrm{Pic}}^{0}_{\underline{A}^{1}/S} \simeq \underline{\mathrm{Pic}}^{0}_{\underline{A}^{0}/S}
\end{cases}
\]

\[
\chi(D) = \tfrac12 D\cdot(D-K) + c
\]

\[
\chi(D) = \ell(D) + \ell(K-D) - \dim H^1(X, \mathcal{L}(D))
\]

\[
\text{\struck{$\chi(\ell$}}\quad \text{\struck{$\chi(D)$}}\quad
\ell(D) + \ell(K-D) \geqslant \tfrac12 D\cdot(D-K) + c
\]

\[
\text{a)}\quad \text{\struck{\ill{}}}\ (D-K)\cdot H \geqslant 0
\]

\[
\text{b)}\quad \tfrac12 D(D-K) + c \geqslant 0
\]

\[
\ell(D + \Delta) > 0
\]

\[
E(X) = \sum_{i=0}^{4} (-1)^i\, b_i(X)
\]

\[
\begin{cases}
b_0(X) = b_4(X) = 1 \\
b_1(X) = b_3(X) = \dim \underline{\mathrm{Pic}}_X \\
E(X) = c_2(X) = \text{self-intersection de la diagonale dans } X\times X.
\end{cases}
\]

\[
\rho = \mathrm{rg}\,(\mathrm{NérSév}_{X/k}),
\]

\[
\boxed{\rho \leqslant b_2}
\]

\[
(*)\qquad \underline{\mathrm{Div}}_{X/S} \simeq \underline{R}^{*}_{X/S} / \underline{O}^{*}_{X} \xrightarrow{\ df/f\ } \underline{R}_{X/S}(\underline{\Omega}^{1}_{X/S}) / \underline{\Omega}^{1}_{X/S}
\]

\[
0 \to \underline{\Omega}^{1}_{X/S} \to \underline{E}_{X/S} \to \underline{\mathrm{Div}}_{X/S} \to 0
\]

\[
0 \to f_{*}(\underline{\Omega}^{1}_{X/S}) \to f_{*}(\underline{E}_{X/S}) \to f_{*}(\underline{\mathrm{Div}}_{X/S}) \xrightarrow{\ \partial\ } R^{1}f_{*}(\underline{\Omega}^{1}_{X/S})
\]

\[
0 \to \underline{\omega}_{X/S} \to f_{*}(\underline{E}_{X/S}) \to f_{*}(\underline{\mathrm{Div}}_{X/S})^{w} \to 0
\]

\[
f_{*}(\underline{\mathrm{Div}}_{X/S})^{w} = \mathrm{Ker}\bigl(\partial\colon f_{*}(\underline{\mathrm{Div}}_{X/S}) \to R^{1}f_{*}(\underline{\Omega}^{1}_{X/S})\bigr)
\]

\[
\underline{P}_{X/S} = f_{*}(\underline{R}^{*}_{X/S}) / f_{*}(\underline{O}^{*}_{X}) \hookrightarrow f_{*}(\underline{\mathrm{Div}}_{X/S})
\]

\[
f_{*}(\underline{\mathrm{Div}}_{X/S}) / \underline{P}_{X/S} \simeq \underline{\mathrm{Pic}}_{X/S}
\]

\[
\begin{aligned}
f_{*}(\underline{\mathrm{Div}}_{X/S})^{w} / \underline{P}_{X/S} &= \underline{\mathrm{Pic}}^{w}_{X/S} \subset \underline{\mathrm{Pic}}_{X/S} \\
&= \mathrm{Ker}\bigl(\underline{\mathrm{Pic}}_{X/S} \to R^{1}f_{*}(\underline{\Omega}^{1}_{X/S})\bigr)
\end{aligned}
\]

\[
  f_*(\underline{R}^{*}_{X/S}) \longrightarrow f_*(\underline{E}_{X/S})
\]

\[
  f \longmapsto (\mathrm{div}\, f, \tfrac{df}{f})
\]

\[
  \mathcal{M}_{X/S} = f_*(\underline{E}_{X/S})/\underline{P}_{X/S}
\]

\[
  0 \to \underline{\omega}_{X/S} \to \mathcal{M}_{X/S} \to \underline{\mathrm{Pic}}_{X/S}^{w} \to 0
\]

\[
  \mathcal{Q} : T \longmapsto \text{classes, à isom.\ près, de faisceaux inversibles}
\]

\[
  H^0(X, \mathcal{O}_X^*) \xrightarrow{\ \varphi \mapsto d\varphi/\varphi\ } H^0(X, \Omega^1_X) \to \mathcal{Q}(T) \to \mathrm{Pic}(X_T) \to H^1(X, \underline{\Omega}^1_{X/S})
\]

\[
  0 \to f_*(\underline{\Omega}^1_X)/\mathrm{Im}\, f_*(\mathcal{O}_X^*) \to \mathcal{M}^{\#}_{X/S} \to \underline{\mathrm{Pic}}^{w}_{X/S} \to 0
\]

\[
  \underline{\mathrm{Pic}}^{w}_{X/S} = \mathrm{Ker}\bigl(\underline{\mathrm{Pic}}_{X/S} \to R^1f_*(\underline{\Omega}^1_{X/S})\bigr)
\]

\[
  \mathrm{Hom}_{G_\alpha}(E, F) \xrightarrow{\ \sim\ } \mathrm{Hom}_{S}(T(E), T(F))
\]

\[
  Z \longrightarrow Z/K, \qquad \mathrm{Hom}_S(Z, Z/K) \simeq G/K
\]

\[
  0 \to \underline{L} \otimes \underline{\Omega}^1_{X/S} \to P^1(\underline{L}) \to \underline{L} \to 0,
\]

\[
  d_L : \underline{L} \to \underline{L} \otimes \Omega^1_{X/S}
\]

\[
  \frac{d_L \varphi_D}{\varphi_D} \in \underline{R}_{U/S}(\underline{L}^{-1} \otimes \underline{L} \otimes \Omega^1_{X/S}) = \underline{R}_{U/S}(\underline{\Omega}^1_{X/S}),
\]

\[
  d_L(\lambda \varphi_D) = \varphi \otimes d(\lambda) + \lambda\, d_L(\varphi) = \varphi \otimes d\lambda + \lambda \varphi \overline{\omega}.
\]

\[
  \overline{\omega} - \frac{df}{f} \in \Gamma(\underline{\Omega}^1_{V/S}),
\]

\[
  f_*(\mathcal{O}_X^*) \xrightarrow{\ \varphi \mapsto d\varphi/\varphi\ } f_*(\underline{\Omega}_{X/S})
\]

\[
  \longrightarrow V(R^1\underline{\mathcal{O}}_{A^*}) \longrightarrow E(A^*) \longrightarrow A^* \longrightarrow 0
\]

\[
  V(R^1\underline{\mathcal{O}}_{A^*}) = W(R^1\underline{\mathcal{O}}_{A^*}{}^{\vee}) = \omega_A
\]

\[
  \widehat{\omega} = \frac{df}{f}, \qquad f\omega = \overline{\omega}, \qquad \omega = \frac{\overline{\omega}}{f}, \qquad \mathrm{res}_x\, \omega,
\]

\[
  \omega_i - \frac{df}{f}, \qquad \omega_i = \frac{df}{f} \ (d \log f)
\]

\[
  \underline{\mathrm{D}}^{\mathrm{lo}}_{X/S} \hookrightarrow \underline{R}_{X/S}(\underline{\Omega}^1_{X/S})/\underline{\Omega}^1_{X/S}
\]

\[
  H^1(S, \underline{\omega}_{X/S}) \simeq \mathrm{Ker}\bigl(H^1(X, \Omega^1_{X/S}) \to H^0(S, R^1f_*(\underline{\Omega}^1_{X/S}))\bigr)
\]

\[
  u : B \to \underline{\mathrm{Pic}}_{X/S}
\]

\[
  \underline{\mathrm{Pic}}^{00}_{X/S} \subset \underline{\mathrm{Pic}}^{w}_{X/S}.
\]

\[
  X \longrightarrow A
\]

\[
  H^0(S, R^1 f_{B*}(\underline{\omega}_{A/S} \otimes_S \underline{\mathcal{O}}_B))
\]

\[
  \underline{H}^{1,1}(A \times_S B) = \underline{H}^{1,1}(A/S) \oplus \underline{H}^{1,1}(B/S)
\]

\[
  + \underline{H}^{1,0}(A/S) \otimes \underline{H}^{0,1}(B/S) + \underline{H}^{0,1}(A/S) \otimes \underline{H}^{1,0}(B/S)
\]

\[
  \underline{W}_{X'_1/S'} \simeq \underline{W}_{X'_2/S'} \quad \text{canoniquement ??}
\]

\[
  \mathrm{Pic}(X) \longrightarrow \mathrm{Pic}(X_m) \qquad \text{inj si } n \geqslant 3,\ \text{bijectif si } n \geqslant 4,
\]

\[
  H^1(X, G) \longrightarrow H^1(Y_{\overline{K}}, G) \quad \text{est injectif.}
\]

\[
  H^1(X, \mathcal{O}_X)^F \to H^1(X, \mathbb{Z}/p\mathbb{Z}) \longrightarrow H^1(Y_{\overline{K}}, \mathcal{O}_{Y_{\overline{K}}})^F
\]

\[
  0 \to \mathcal{O}_X \xrightarrow{\ F\ } \mathcal{O}_X \to \mathcal{O}_X/\mathcal{O}_X^p \to 0
\]

\[
  1 \to \mathcal{O}_X^* \xrightarrow{\ F\ } \mathcal{O}_X^* \to \mathcal{O}_X^*/\mathcal{O}_X^{*p} \to 1
\]

\[
  H^1(X, \alpha_p) = H^1(X, \mathcal{O}_X)^{1-F} \simeq H^0(X, \mathcal{O}_X/\mathcal{O}_X^p) \xrightarrow{\ df\ } H^0(X, \widetilde{\underline{\Omega}}^1_X)
\]

\[
  H^1(X, \mu_p) = H^1(X, \mathcal{O}_X)^{1-F} \simeq H^0(X, \mathcal{O}_X/\mathcal{O}_X^*) \xrightarrow{\ df/f\ } H^0(X, \widetilde{\underline{\Omega}}^1_X)
\]

\[
  \mathrm{Pic}(X) \longrightarrow \mathrm{Pic}\, X[t, t^{-1}]
\]

\[
f^*i_*(\underline{F})\simeq i'_*g^*(\underline{F})\simeq i'_*i'^*(\underline{E}')
\simeq\underline{E}' , \quad \text{OK.}
\]

\[
f^*(\underline{E})\simeq\underline{E}' \Longleftarrow
i'^*f^*(\underline{E})\simeq i'^*(\underline{E}')
\]

\[
\boxed{A^*=A_f^*\cap\bigcap_i A^*_{\mathfrak{p}_i}}
\]

\[
a^n+f^kc_1a^{n-1}+f^{2k}c_2a^{n-2}+\cdots+f^{nk}c_n=0
\]

\[
  t_i = \varepsilon_i f_i^{\,n_i}, \qquad \varepsilon_i \in A_i^{*},
\]

\[
  t_i = \varepsilon_i f_i^{\,n_i} \qquad (\varepsilon_i \in A_i^{*},\
  n_i \in \mathbb{Z}).
\]

\[
  t_{ij} = \varepsilon_{ij} f_{ij}^{\,n_i} = \varepsilon_{ji} f_{ji}^{\,n_j},
\]

\[
  f_{ij}^{\,(n_i - n_j)} = \varepsilon_{ji}/\varepsilon_{ij},
\]

\[
  \mathcal{S}(\underline{L}_1, X_1, Y_1) \simeq
  \mathcal{S}(\underline{L}/X/Y) \times_Y Y_1,
\]

\[
  K \rightrightarrows M, \qquad L \rightrightarrows N, \qquad
  L \rightrightarrows P
\]

\[
  K \rightrightarrows M, \qquad L \rightrightarrows N, \qquad
  \text{i.e.} \quad K \times M \rightrightarrows L \times N .
\]

\[
  \underline{\mathrm{Pic}}_{X/k} \longrightarrow
  \prod_i \underline{\mathrm{Pic}}_{X_i/k}
\]

\[
  \underline{\mathrm{Pic}}_{X/k} \longrightarrow
  \underline{\mathrm{Pic}}_{X_1/k} \times \underline{\mathrm{Pic}}_{X_2/k}
\]

\[
  X \to Y' \to Y
\]

\[
  X \to Y' \to Y ;
\]

\[
  X \xrightarrow{\ \varphi\ } \Sigma \xrightarrow{\ \psi\ } S,
\]

\[
  0 \to \mathrm{Pic}(\Sigma) \to \mathrm{Pic}(X) \to \mathrm{Pic}'(X/S)
\]

\[
  \mathrm{Pic}'(X/S) \simeq \mathrm{Pic}(X)/\mathrm{Pic}(\Sigma).
\]

\[
  \mathrm{Pic}'(X/S) \to \mathrm{Pic}'(X'/S') \rightrightarrows
  \mathrm{Pic}'(X''/S'')
\]

\[
  \mathrm{Pic}'(X/S) \to \mathrm{Pic}^{*}(X/S)
\]

\[
  \mathrm{Pic}(X/S) \simeq \mathrm{Pic}(X)/\mathrm{Pic}(\Sigma).
\]

\[
  E_2^{pq} = H^p(S, R^q_! f_U(\mu_n)) \Longrightarrow H^*_{\mathrm{prop}}(U, \mu_n),
\]

\[
  R^q_! f_U(\mu_n) =
  \begin{cases}
    0 & \text{si } q \neq 1, 2 \\
    {}_n E & \text{si } q = 1 \\
    (\mathbb{Z}/n\mathbb{Z})_S & \text{si } q = 2
  \end{cases}
\]

\[
  \begin{array}{ccc}
    E_2^{0,2} & \longrightarrow & E_2^{2,1} \\
    \| & & \| \\
    H^0(S, \mathbb{Z}/n\mathbb{Z}) & & H^2(S, {}_n E)
  \end{array}
\]

\[
  \xi_n \in H^2(S, {}_n E).
\]

\[
  \boxed{\xi(\ell) \in H^2(S, T_\ell(E))}
\]

\[
  0 \to T \to E \to J \to 0, \qquad T = \mathbb{G}_{m,S}^I / \mathbb{G}_{m,S}.
\]

\[
  X \amalg (Y \times_S Y) \rightrightarrows X
  \qquad (\alpha : Y \hookrightarrow X),
\]

\[
  E = \underline{\mathrm{Pic}}^0_{Z/S}.
\]

\[
  P = \underline{\mathrm{Pic}}^1_{Z/S}
\]

\[
  \boxed{\eta \in H^1(S, E)},
\]

\[
  \boxed{\xi_n = \partial_n(\eta)},
\]

\[
  H^1(S, E) \longrightarrow H^2(S, {}_n E)
\]

\[
  0 \to {}_n E \to E \xrightarrow{\;n_E\;} E \to 0 .
\]

\[
  H^0(S, J) \to H^1(S, T) \to H^1(S, E) \to H^1(S, J)
\]

\[
  (*) \qquad 0 \to \mathbb{G}_{m,S} \to \prod_{Y/S} \mathbb{G}_m \to T \to 0
\]

\[
  \begin{aligned}
  0 &\to H^0(S, \mathcal{O}_S)^* \to H^0(Y, \mathcal{O}_Y)^* \to H^0(S, T) \to \operatorname{Pic}(S) \to \operatorname{Pic}(Y) \\
    &\to H^1(S, T) \to \mathrm{Br}'(S) \to \mathrm{Br}'(Y),
  \end{aligned}
\]

\[
  \text{\struck{$H^1(S,T) =$}} \quad H^1(S, T) \simeq \operatorname{Pic}(Y) / \operatorname{Im} \operatorname{Pic}(S),
\]

\[
  \operatorname{Pic}(Y) \simeq \operatorname{Pic}(S)^I
\]

\[
  H^1(S, T) \simeq \operatorname{Pic}(S)^I / \operatorname{Im} \operatorname{Pic}(S)
\]

\[
  0 \to \underline{\mathcal{O}}_C^* \to \underline{\widetilde{\mathcal{O}}}_C^* \to \prod_p A_p^* \to 0
  \qquad p \text{ les pts doubles}
\]

\[
  0 \to \mathbb{G}_m \to \prod_{C_i} \mathbb{G}_m \to \prod_{p_\alpha} A_p \to \operatorname{Pic}_{C/k} \to \prod \operatorname{Pic}_{\widetilde{C}_i/k} \to 0
\]

\[
  G = \Bigl(\prod_{p_\alpha} A_{p_\alpha}\Bigr) \Big/ \bigl(\textstyle\prod \mathbb{G}_m / \mathbb{G}_m\bigr)
\]

\[
  \dim G = \sum \mu_\alpha - c_0 + 1
\]

\[
  \mu_\alpha = 1 + \delta_\alpha, 
\]

\[
  \mu_\alpha = \mu'_\alpha + \mu''_\alpha ,
\]

\[
  \text{\struck{Dim}}\ \dim G_{\mathrm{mult}} = \sum \mu'_\alpha - c_0 + 1 = h_{\underline{1}}
\]

\[
  \prod \pi_1(\widetilde{C}_i) \to \pi_1(C) \;]
\]

\[
  \text{\struck{Dim}}\ \dim G_{\mathrm{add}} = \sum \mu''_\alpha
\]

\[
  \dim G_{\mathrm{mult}} = \sum (\nu_\alpha - 1) - c_0 + 1
  = \sum_i (\text{nb des pts } P_\alpha \text{ sur } C_i - 1) - \cdots
\]

\[
  \begin{aligned}
  g = \dim \operatorname{Pic}_{C/k} &= \sum g_i + \Bigl(\sum_\alpha \mu_\alpha - c_0 + 1\Bigr) \\
  &= \sum g_i + \rho + \sigma
  \end{aligned}
\]

\[
  g' + g'' = g \qquad (g' g'' > 0)
\]

\[
  \sum_{g_i \geq 1} (3g_i - 3 + \nu_i)
  + \sum_{g_i = 1} \underbrace{(1 + \nu_i - 1)}_{3g_i - 3 + \nu_i}
  + \sum_{\substack{g_i = 0 \\ \nu_i \geq 3}} \underbrace{\nu_i - 3}_{3g_i - 3 + \nu_i}
\]

\[
  3 \sum g_i - 3c_0 + \sum \nu_i
  \;\text{\struck{$+$}}\; {}^{\lambda}\!\!\sum_{\substack{g_i = 0 \\ \nu_i = 2}} (3 - \nu_i)
\]

\[
  \sum g_i = g - \rho \qquad (\rho = \text{nombre cycles combinatoires})
\]

\[
  \sum_i \nu_i = \sum_\alpha (\mu_\alpha)
  = \text{\struck{$\sum$}}\ \underbrace{(\mu_\alpha - 1)}_{c_0 - 1 + \rho} + c_1
\]

\[
  \boxed{\sum \nu_i = c_0 - 1 + \rho + c_1}
  \qquad \rho = g - \sum g_i
\]

\[
  \begin{aligned}
  &3g - 3\rho - 3c_0 + \text{\struck{$c_0$}}\ \underbrace{\ -1 + \rho}\ + c_1 + \lambda \\
  &3g - 2\rho - 2c_0 + c_1 + \lambda - 1 \\
  &3g - 3\ \bigl[2(c_0 + \rho) - c_1 - \lambda - 2\bigr]
  \end{aligned}
\]

\[
  \text{\struck{$k$}} \qquad -6 + 2 - 1 = -5 \qquad 9 - 5 = 4
\]

\[
    g : X \to \mathbb{P}^N_S
  \]

\[
    g_*(\underline{\mathcal{O}}(1)) \to f_*(\underline{L}) = E
  \]

\[
    \varphi : \mathbb{P}^N_S \to P(E)
  \]

\[
    g : X \to P
  \]

\[
    \varphi : \mathbb{P}^N_\xi \xrightarrow{\sim} P_\xi
  \]

\[
    g : X \to \mathbb{P}^N_S
  \]

\[
\mathcal{P} \longrightarrow \mathcal{J}
\]

\[
\cdot \to \mathrm{Ker} \to \underline{H}^{0}_{f}(\underline{L}^{Y})^{\vee} \to \underline{\mathcal{O}}_{Y} \to 0
\]

\[
0 \to \underline{\mathcal{O}}_{S} \to \underline{H}^{0}_{f}(\underline{L}) \to \underline{H}^{0}_{f}(\underline{L}|Y)
\]

\[
\underline{J}\,\underline{L} \to \underline{L} \to \underline{L}|Y \to 0
\qquad\qquad \underline{L} = \underline{J}^{-1}
\]

\[
0 \to \underline{\mathcal{O}}_{X} \to \underline{L} \to \underline{L}|Y \to 0
\]

\[
0 \to \underline{\mathcal{O}}_{S} \to \underline{H}^{0}_{f}(\underline{L}) \to \underline{H}^{0}_{f}(\underline{L}|Y) \to \underline{H}^{1}_{f}(\underline{\mathcal{O}}_{X})
\]

\[
\to \underline{H}^{1}_{f}(\underline{L}) \longrightarrow \underline{H}^{1}_{f}(\underline{L}|Y) \to \underline{H}^{1}_{f}(\underline{\mathcal{O}}_{X})
\]

\[
\mathrm{div} f = \sum_{\substack{V \text{ div.} \\ \text{pr}/k}} v_{V}(f) \cdot V
\]

\[
\underline{\underline{\text{Prépic}}}_{X/S}(S') = \text{Prépic}(X'/S') = H^{1}(X', \underline{\mathcal{O}}_{X'}^{*})
\qquad \text{où } X' = X \times_{S} S'
\]

\[
\text{Prépic}'(X/S) = \Gamma(S, \underline{H}^{1}_{X/S}(\underline{\mathcal{O}}_{X}^{*}))
\]

\[
\underline{\underline{\text{Prépic}}}'_{X/S}(S') = \Gamma(S', \underline{H}^{1}_{X'/S'}(\underline{\mathcal{O}}_{X'}^{*}))
\qquad \text{où } X' = X \times_{S} S'
\]

\[
\text{Prépic}'(X/S) \leftarrow \text{Prépic}(X/S)/\mathrm{Pic\,abs}(S)
\]

\[
0 \to \mathrm{Pic}(S) \to \mathrm{Pic}(X) \to \text{Prépic}'(X/S)
\]

\[
0 \to H^{1}(S, \underline{H}^{0}(\underline{\mathcal{O}}_{X}^{*})) \to H^{1}(X, \underline{\mathcal{O}}_{X}^{*}) \to H^{0}(S, \underline{H}^{1}(\underline{\mathcal{O}}_{X}^{*})) \to H^{2}(S, \underline{H}^{0}(\underline{\mathcal{O}}_{X}^{*}))
\]

\[
H^{2}(S, \underline{H}^{0}(\underline{\mathcal{O}}_{X}^{*})) \to H^{2}(X, \underline{\mathcal{O}}_{X}^{*})
\]

\[
\text{\struck{$\mathrm{Pic}$}}\ \text{Prépic}'(X/S) \to \text{Prépic}'(X'/S')
\]

\[
\to \text{Prépic}'(X/S) \to \text{Prépic}'(X'/S') \rightrightarrows \text{Prépic}'(X''/S'')
\]

\[
\underline{\underline{\text{Prépic}}}_{X/S} \xrightarrow{\alpha} \underline{\underline{\text{Prépic}}}'_{X/S} \xrightarrow{\beta} \underline{\underline{\mathrm{Pic}}}_{X/S}
\]

\[
H^{0}(S, \underline{H}^{0}_{f}(\underline{L})^{*}/\underline{\mathcal{O}}_{S}^{*})
\]

\[
\underline{\underline{\mathrm{Div}}}^{L}_{X/S}(S') = H^{0}(S', \underline{H}^{0}_{f'}(\underline{L}')^{*}/\underline{\mathcal{O}}_{S'}^{*})
\]

\[
  \mathbf{Div}_{X/S} \simeq
  \mathbf{Div}^{\underline{\Lambda}}_{\mathbf{Pic}_{X/S} \times_S X \,/\, \mathbf{Pic}_{X/S}} .
\]

\[
  \underline{H}^0_f(\underline{L} \otimes_{\underline{O}_S} \underline{M})
  \simeq \mathbf{Hom}_{\mathcal{O}}(\underline{E}, \underline{M})
\]

\[
  \mathbf{Div}^{\underline{L}}_{X/S} \simeq \underline{P}(\underline{E}),
\]

\[
  \mathrm{Pic}(C - a) \simeq \mathrm{Pic}(X)/\mathbb{Z}.\xi
\]

\[
  C_a - a \longrightarrow C - a ,
\]

\[
  \mathrm{Pic}(C - a) \longrightarrow \mathrm{Pic}(C_a - a)
  \simeq \varinjlim_{U \ni a} \mathrm{Pic}(U - a)
\]

\[
  \mathrm{Pic}(U - a) \simeq \mathrm{Pic}(C - a)/\mathrm{Im}\, \mathbb{Z}^{(I)}
\]

\[
  D|V \sim 0 \quad \text{dans } V.
\]

\[
  D = (h'_* q)_* (1_{X'}) = h'_*(q_*.1_{X'})
\]

\[
  q_*(1_{X'}) = f'^*(\xi_{E'})
\]

\[
  q_*(1_{X'}) = f'^* h^*(\xi_E) = h'^* f^*(\xi_E)
\]

\[
  D = h'_* h'^*(f^*(\xi_E)) = f^*(\xi_E)\, h'_*(1_{E'})
\]

\[
  D = d\, f^*(\xi_E), -
\]

\[
  D|V = d\,(f^*(\xi_E)|V) = 0 \qquad \text{cqfd.}
\]

\[
  \boxed{\mathrm{Pic}(C - a) \simeq \mathrm{Pic}(C_a - a) \simeq \mathrm{Pic}(X)/\mathbb{Z}\xi}
\]

\[
  S = \text{\struck{$\mathrm{Spec}$}}\ C_a = \mathrm{Spec}\, \mathcal{O}_{C,a}, \qquad
  \hat{S} = \mathrm{Spec}\, \widehat{\mathcal{O}_{C,a}}
\]

\[
  \mathrm{Pic}(S - a) \longrightarrow \mathrm{Pic}(\hat{S} - \hat{a})
\]

\[
  \mathrm{Cl}(S) \longrightarrow \mathrm{Cl}(\hat{S})
\]

\[
  \mathrm{Pic}(\hat{S} - \hat{a}) \simeq \mathrm{Pic}(\widetilde{\hat{S}} - X)
  \simeq \mathrm{Pic}(\widetilde{\hat{S}})/\mathbb{Z}\hat{\eta}
\]

\[
  \widetilde{\hat{S}} \simeq \tilde{S} \times_S \hat{S} \simeq \widehat{\tilde{S}}
  \qquad \text{(complété formel de $\tilde{S}$ le long de $X$)}
\]

\[
  \mathrm{Pic}(\widetilde{\hat{S}}) \simeq \mathrm{Pic}(\widehat{\tilde{S}}) ;
\]

\[
  \mathrm{Pic}(S - a) \longrightarrow \mathrm{Pic}(\hat{S} - \hat{a})
\]

\[
  \mathrm{Pic}(E) \longrightarrow \mathrm{Pic}(\hat{E})
\]

\[
  \mathrm{Pic}(\hat{E}) = \varprojlim \mathrm{Pic}(E_n)
\]

\[
  E_n = (X, \mathcal{O}_E/\mathcal{J}^{n+1})
\]

\[
  \mathcal{J}/\mathcal{J}^2 = \mathcal{O}_X(1)
\]

\[
  \mathrm{Pic}(E) \xleftarrow{\ \sim\ } \mathrm{Pic}(X)
\]

\[
  H^1(X, \mathcal{J}^n/\mathcal{J}^{n+1}) \simeq H^1(X, \mathcal{O}_X(n)) .
\]

\[
  \mathrm{Pic}(S - a) \to \mathrm{Pic}(\tilde{S} - \tilde{a}) \to \mathrm{Pic}(\hat{S} - \hat{a}) .
\]

\[
  T_t(G) \simeq T(G)_t \qquad \text{où } T(G) = G/A.
\]

\[
  \sigma(s) \geq \mathrm{rang}_{\mathcal{O}}[Q_s : k(s)] \geq
  \mathrm{rang}_{\mathcal{O}}[Q_{s'} : k(s')] = \sigma(s')
\]

\[
  \mathrm{long}\, H^1(\bar{X}_s, \mathbb{Z}/p^n)
  + \mathrm{long}\, H^1(\bar{X}_s, \mu_{p^n})
\]

\[
  + \;\struck{\ill{}}\ \mathrm{long}_{W_n^\sigma[F,V]}\, H^1(\bar{X}_s, W_{n,n})
\]

\[
  \mathrm{Hom}(\pi_1(\bar{X}_s), \mathbb{Z}/p^n) , \qquad
  {}_{p^n}\mathrm{Pic}(\bar{X}_s) ,
\]

\[
  \mathrm{Ker}\bigl(H^1(\bar{X}_s, W_n) \xrightarrow{F^n} H^1(\bar{X}_s, W_n)\bigr)
\]

\[
  U^2 = G_0
\]

\[
  X'_\eta \simeq X_\eta = V_\eta .
\]

\[
  Z = (Z \cap V = Z') \cup (Z - Z'),
\]

\[
  S = \operatorname{Spec} k[t], \qquad X = \operatorname{Proj} k[t, x, y, z]/(xy - tz^2)
\]

\[
  (*) \qquad H^1(X', \mathcal{O}^*_{X'}) \to H^1(X'_0, \mathcal{O}_{X'_0})
\]

\[
  L_{D_1} | Y_1 \simeq - L_{D_2} | Y_1 .
\]

\[
  0 \to \mathbb{Z} \xrightarrow{\ \alpha\ } H^1(X, \mathcal{O}^*_X) \xrightarrow{\ \beta\ } \mathbb{Z} \to 0
\]

\[
  H^1(X, \mathcal{O}^*_X) \to H^1(X_a, \mathcal{O}^*_{X_a}) \simeq \mathbb{Z} \times \mathbb{Z}
\]

\[
  H^1(X|U, \mathcal{O}^*_{X|U}) = H^1(\mathbf{P}^1_U, \mathcal{O}^*_{\mathbf{P}^1_U})
  = H^1(U, \mathcal{O}^*_U) \times \mathbb{Z} \simeq \mathbb{Z}
\]

\[
  S_I = \bigcup_{i \in I} S_i , \qquad S_i \cap S_j = U \times \{I\}
\]

\[
  \mathfrak{P} = \coprod_{n \in \mathbb{Z}} S_{I_n} .
\]

\[
  \mathfrak{P} = \bigcup_{(p, q) \in \mathbb{Z} \times \mathbb{Z}} S_{(p,q)}
  \quad \text{où chaque } S_{(p,q)} \simeq S
\]

\[
  H^1(\hat{C}, \mathcal{O}^*_{\hat{C}}) \to H^1(\hat{C}_X, \mathcal{O}^*_{\hat{C}_X})
\]

\[
  \varepsilon^*(\xi) = (\varphi g)^*(\eta) \, \varepsilon^*(\mathcal{O}_{\hat{C}_X}(n)) = \eta ,
\]

\[
  \boxed{H^1(\hat{C}, \mathcal{O}^*_{\hat{C}}) \simeq \mathbb{Z}, \quad \text{engendré par } \operatorname{cl}(\mathcal{O}_{\hat{C}}(1))}
\]

\[
  X = \operatorname{Proj}\bigl(k[x, y, z, t]/(x^2 + y^2 + z^2 - \lambda t^2)\bigr)
  \quad \text{sur} \quad S = \operatorname{Spec}(\struck{k}\,\ill{})
\]

\[
A[x,y,z]/xy(x+y+tz) = \mathcal{S}
\]

\[
\begin{array}{ccc}
(x) & (y) & (x+y+tz) \\
\mathfrak{p}_1 & \mathfrak{p}_2 & \mathfrak{p}_3 \\
\underline{I}_1 & \underline{I}_2 & \underline{I}_3 \\
X_1 & X_2 & X_3
\end{array}
\]

\[
\bigvee_{1 \leqslant i \leqslant 3} X_i = X .
\]

\[
H^1(X, \underline{O}_X) \simeq H^1(X_1 \sqcup X_2 \sqcup X_3 / X, \underline{G}_a) \simeq
\]

\begin{align*}
\operatorname{Pic}(X_y) &= R^1 f(\underline{O}_X^{\bullet})_y
  = \varinjlim_{U \ni y} \operatorname{Pic}(f^{-1}(U)) ,\\
\operatorname{Pic}(\hat{X}_y) &= \varprojlim \operatorname{Pic}(X_n) ,
\end{align*}

\[
\operatorname{Pic}(X_y) \xrightarrow{\ u\ } \operatorname{Pic}(\hat{X}_y)
\xrightarrow{\ v_n\ } \operatorname{Pic}(X_n)
\]

\[
\underline{\mathrm{Pic}}_{X_m/\underline{C}} \to \underline{\mathrm{Pic}}_{X_n/\underline{C}}
\]

\[
\operatorname{Ker}\underline{\Pi}_0(\varphi)
\text{\struck{$= \operatorname{Ker}\underline{\Pi}_0(i)$}}
\simeq \bigl(H'/H'^0\bigr) \cap \bigl(H/H^0\bigr)^0 .
\]

\[
0 \to H^0 \to H/G \to \operatorname{Coker}\underline{\Pi}_0(\varphi) \to 0 .
\]

\[
G_{n,m}(k) = \operatorname{Im}(G_m(k) \to G_n(k))
\]

\[
\bigl(\underline{\mathrm{Pic}}_{X_n/k}\bigr) .
\]

\[
\operatorname{Pic}(X_n) \xrightarrow{\ \sim\ } \underline{\operatorname{Pic}}_{X_n/k}(k)
\]

\[
\text{\struck{$\operatorname{Pic}(X_m) \longrightarrow \operatorname{Pic}(X_n)$}}
\]

\[
0 \longrightarrow \operatorname{Pic}(X_n) \longrightarrow \underline{\operatorname{Pic}}_{X_n/k}(k) \longrightarrow \check{H}^2(k'/k, M_n)
\]

\[
\check{H}^*(k'/k, M_m) \xrightarrow{\ \sim\ } \check{H}^*(k'/k, M_n)
\]

\[
\varinjlim_{U} H^*(U, F) \longrightarrow H^*(Y, F|Y)
\]

\[
R^*f_*(F)_y \simeq H^*(X_y, F|X_y)
\]

\[
R^if_*(F)_y^{\wedge} \longrightarrow \varprojlim_n H^i(X_n, F_n)
\]

\[
R^if_*(F)_y \longrightarrow \varprojlim_n H^i(X_n, F_n)
\]

\[
0 \longrightarrow \mathbb{Z}_{X_n} \xrightarrow{\ \alpha\ } \mathcal{O}_{X_n} \xrightarrow{\ \beta\ } \mathcal{O}^*_{X_n} \longrightarrow 0
\]

\[
\begin{aligned}
\cdots \longrightarrow H^i(X_n, \mathbb{Z}) &\xrightarrow{\ \alpha^i\ } H^i(X_n, \mathcal{O}_{X_n}) \xrightarrow{\ \beta^i\ } H^i(X_n, \mathcal{O}^*_{X_n}) \\
&\xrightarrow{\ \partial^i\ } H^{i+1}(X_n, \mathbb{Z}) \xrightarrow{\ \alpha^{i+1}\ } H^{i+1}(X_n, \mathcal{O}_{X_n})
\end{aligned}
\]

\[
\begin{aligned}
H^i(X_0, \mathbb{Z}) &\xrightarrow{\ \alpha^{(i)}_\infty\ } R^if_*(\mathcal{O}_X)_y^{\wedge} \xrightarrow{\ \beta^{(i)}_\infty\ } \varprojlim_n H^i(X_n, \mathcal{O}^*_{X_n}) \\
&\xrightarrow{\ \partial^{(i)}_\infty\ } H^{i+1}(X_0, \mathbb{Z}) \xrightarrow{\ \alpha^{(i+1)}_\infty\ } R^{i+1}f_*(\mathcal{O}_X)_y^{\wedge}
\end{aligned}
\]

\[
0 \longrightarrow \mathbb{Z}_X \longrightarrow \mathcal{O}_X \longrightarrow \mathcal{O}^*_X \longrightarrow 0
\]

\[
\varinjlim_{U \ni y} H^i(f^{-1}(U), \mathcal{O}^*) = R^if_*(\mathcal{O}^*_X)_y \longrightarrow \varprojlim_n H^i(X_n, \mathcal{O}^*_{X_n})
\]

\[
R^if_*(\mathcal{O}^*_X)_y \longrightarrow \varprojlim_n H^i(X_n, \mathcal{O}^*_{X_n})
\]

\[
\operatorname{Pic}(\hat{X}_y) = \varprojlim_n \operatorname{Pic}(X_n).
\]

\[
\varphi : \operatorname{Pic}(X_y) \longrightarrow \operatorname{Pic}(\hat{X}_y)
\]

\[
\mathrm{Pic}(\hat{X}) \simeq \varprojlim \mathrm{Pic}(X_n).
\]

\[
  \varphi : \mathrm{Pic}(X) \to \mathrm{Pic}(\hat{X})
  \]

\[
\mathrm{Pic}(Y_y) \longrightarrow \mathrm{Pic}(\tilde{Y}_{\tilde{y}}) \quad \text{est un \add{\emph{iso}}\struck{\emph{monomorphisme}}morphisme.}
\]

\[
\Gamma(Y_1) \to \Gamma(Y'_1) \quad \text{est un \uncertain{isom.}}
\]

\[
\mathrm{Pic}(Y') \to \mathrm{Pic}(\tilde{Y}') \quad\text{est \emph{injectif}.}
\]

\[
\mathrm{Pic}(\text{\struck{$X$}}\,Y') \simeq \mathrm{Pic}(X') \simeq \mathrm{Pic}(X)/I \simeq \mathbf{Z}^{I}
\]

\[
\mathrm{Pic}(\tilde{Y}') \simeq \mathrm{Pic}(\tilde{X}') \simeq \mathrm{Pic}(\tilde{X})/I \simeq \mathbf{Z}^{I}
\]

\[
  \mathrm{Pic}(X) \xrightarrow{\ \sim\ } \mathrm{Pic}(\hat{X}) \ ??
  \]

\[
  \mathrm{Pic}(Y - \{y\}) \xrightarrow{\ \sim\ } \mathrm{Pic}(\hat{Y} - \{\hat{y}\})
  \]

\[
\mathrm{N\acute{e}r}'(X/S) = \mathrm{Pic}(X/S)/\mathrm{Pic}^{\tau}(X/S).
\]

\[
\mathrm{N\acute{e}r}'(X/S) \longrightarrow \mathrm{N\acute{e}r}'(X'/S') \quad\text{est \emph{injectif}.}
\]

\[
(\mathrm{Sch})^{\circ}_{/S} \longrightarrow (\mathrm{Ens}),
\]

\[
\underline{\mathrm{N\acute{e}r}}_{X/S} = \underline{\mathrm{Pic}}^{*}_{X/S}/\underline{\mathrm{Pic}}^{\tau}_{X/S}
\]

\[
\underline{\mathrm{N\acute{e}r}}''_{X/S}(S) = \mathrm{N\acute{e}r}''(X/S),
\]

\[
\text{(\ill{})}\quad \underline{\mathrm{N\acute{e}r}}''_{X/S}(T) \simeq \mathrm{N\acute{e}r}''(X_T/T) \quad \text{(isom.\ can.)},
\]

\[
0 \to \mathrm{Pic}(Y) \to \mathrm{Pic}(X) \to \mathrm{Pic}(X/Y)
\qquad \mathrm{Pic}(X/Y) = \Gamma(\underline{\mathrm{Pic}}_{X/Y}/Y)
\]

\[
0 \to \underline{\mathrm{Pic}}_{Y/S} \to \underline{\mathrm{Pic}}_{X/S} \to \textstyle\prod_{Y/S} \underline{\mathrm{Pic}}_{X/Y} ,
\]

\[
0 \to \underline{\mathrm{Pic}}_{Y/S} \to \underline{\mathrm{Pic}}_{X/S} \to \textstyle\prod_{Y/S} \underline{\mathrm{Pic}}_{X/Y}
\]

\[
\underline{\mathrm{Pic}}_{X/S} \simeq \underline{\mathrm{Pic}}_{Y/S} \times \textstyle\prod_{Y/S} \underline{\mathrm{Pic}}_{X/Y}
\]

\[
\underline{\mathrm{Pic}}_{X/S} \simeq \underline{\mathrm{Pic}}_{Y/S} \times \underline{\mathrm{Pic}}_{Z/S} \times \underline{\mathrm{Hom}}_{\text{pointés}}(Y, \underline{\mathrm{Pic}}_{Z/S}) \;]
\]

\[
\underline{\mathrm{Pic}}_{X/S} \simeq \underline{\mathrm{Pic}}_{Y/S} \times \textstyle\prod_{Y/S} \underline{\mathrm{Pic}}_{X/Y} ,
\]

\[
\underline{\mathrm{Pic}}_{X/S} \simeq \underline{\mathrm{Pic}}_{Y/S} \times \textstyle\prod_{Y/S} \underline{\mathrm{Pic}}_{X/Y} ,
\qquad
\textstyle\prod_{Y/S} \underline{\mathrm{Pic}}_{(A' \times B)/\Gamma} \simeq \underline{\mathrm{Hom}}_{S,\Gamma}(B, \underline{\mathrm{Pic}}_{A/S})
\]

\[
\underline{\mathrm{Hom}}_{S,\Gamma}(B, A')^{(\tau)} \simeq A'^{\Gamma} ,
\]

\[
\underline{\mathrm{Pic}}^{\tau}_{X/S} \simeq \underline{\mathrm{Pic}}^{\tau}_{Y/S} \times A'^{\Gamma}
\]

\[
\underline{\mathrm{Pic}}^{\tau}_{X/S} \simeq Y' \times {}_2A'
\]

\[
\dim B + \dim A
\]

\[
0 \to \underline{\mathrm{Pic}}_{A/S} \to \underline{\mathrm{Hom}}_S(B, \underline{\mathrm{Pic}}_{A/S}) \to \underline{\mathrm{Hom}}_{S\text{-gr}}(A, \underline{\mathrm{Pic}}^{\tau}_{B/S}) \to 0
\]

\[
0 \to \underline{\mathrm{Pic}}_{A/S}^{\Gamma} \to \underline{\mathrm{Hom}}_{S,\Gamma}(B, \underline{\mathrm{Pic}}_{A/S}) \to \underline{\mathrm{Hom}}_{S\text{-gr}}(A, \underline{\mathrm{Pic}}^{\tau}_{B/S})^{\Gamma}
\]

\[
\underline{\mathrm{Hom}}_{S\text{-gr}}(A, \underline{\mathrm{Pic}}^{\tau}_{B/S})^{\Gamma} = 0
\]

\[
\underline{\mathrm{Pic}}_{A/S}^{\Gamma} \simeq \underline{\mathrm{Hom}}_{S,\Gamma}(B, \underline{\mathrm{Pic}}_{A/S}) .
\]

\[
(\ast)\qquad
\begin{cases}
\underline{\mathrm{Pic}}_{X/S} \simeq \underline{\mathrm{Pic}}_{Y/S} \times \underline{\mathrm{Pic}}_{A/S}^{\Gamma} \\
\underline{\mathrm{Pic}}^{\tau}_{X/S} \simeq \underline{\mathrm{Pic}}^{\tau}_{Y/S} \times A'^{\Gamma}
\end{cases}
\]

\[
p_*(\underline{\Omega}^1_{X/S}) = q_*(\underline{\Omega}^1_{A \times_S B})^{\Gamma} = \pi_{A*}(\underline{\Omega}^1_{A/S})^{\Gamma} \times \pi_{B*}(\underline{\Omega}^1_{B/S})^{\Gamma}
\]

\[
\text{a)}\qquad \omega^1_{X/S} \simeq \omega^1_{B/S} \times {\omega^1_{A/S}}^{\Gamma}
\]

\[
\begin{aligned}
&\text{a}')\quad H^0(X, \Omega^1_X) \simeq H^0(B, \Omega^1_B) \times H^0(A, \Omega^1_A)^{\Gamma} \\
&\text{b}')\quad H^1(X, \mathcal{O}_X) \simeq H^1(Y, \mathcal{O}_Y) \times \underbrace{H^1(A, \mathcal{O}_A)^{\Gamma}}_{t_{A'}^{\Gamma}}
\end{aligned}
\]

\[
\operatorname{rang} H^1(X, \mathcal{O}_X) - \operatorname{rang} H^0(X, \Omega^1_{X/k}) = \dim t_{A'}^{\Gamma} - \dim (t_A)'^{\Gamma}
\]

\[
A = X/F , \qquad B = X/G , \qquad X \xrightarrow{\ \alpha\ } A , \quad X \xrightarrow{\ \beta\ } B \quad \text{hom.\ can.}
\]

\[
\mathrm{Hom}(X_1, X_1) \otimes \mathbb{Q} \simeq \mathrm{Hom}(A_1, B_1) \otimes \mathbb{Q} ,
\]

\[
C = A \times B' \qquad B' \text{ dual de } B
\]

\[
\underline{\mathrm{NS}}_{C/Y} \simeq \underline{\mathrm{NS}}_{A/Y} \times \underline{\mathrm{NS}}_{B/Y} \times \underline{\mathrm{Hom}}_{Y\text{-gr}}(A, B) .
\]

\[
{}_pM = \mathrm{Hom}(\mathbb{Z}/p\mathbb{Z}, M) .
\]

\[
T_p(M) = \mathrm{Hom}(M, \mathbb{Q}_p/\mathbb{Z}_p) = \varprojlim_n \mathrm{Hom}({}_{p^n}M, \mathbb{Q}_p/\mathbb{Z}_p)
\]

\[
T^p(M) = \varprojlim_n {}_{p^n}M
\]

\[
T^p(M) = \mathrm{Hom}(T_p(M), \mathbb{Z}_p)
\]

\[
\Updownarrow
\]

\[
\check V \xrightarrow{\ \mu\ } H^1(X_0, \underline{\mathcal{T}}_{X_0}) \xrightarrow{\ u\ } H^1(X_0, \mathcal{O}_{X_0}) \otimes t_{B_0} \simeq H^1(X_0, \mathcal{O}_{X_0}) \otimes H^1(A_0, \mathcal{O}_{A_0})
\]

\[
v : H^1(A_0, \underline{\mathcal{T}}_{A_0}) \simeq t_{A_0} \otimes H^1(A_0, \mathcal{O}_{A_0}) \to H^1(X_0, \mathcal{O}_{X_0}) \otimes H^1(A_0, \mathcal{O}_{A_0})
\]

\[
\Downarrow
\]

\[
H^1(X_0, \mathcal{O}^*_{X_0}) \xrightarrow{\ \partial\ } \mathrm{Hom}\bigl(H^1(X_0, \underline{\mathcal{T}}_{X_0}), H^2(X_0, \mathcal{O}_{X_0})\bigr) \xrightarrow[\ \rho\ ]{} \mathrm{Hom}\bigl(\check V, H^2(X_0, \mathcal{O}_{X_0})\bigr)
\]

\[
\rho\partial(x) = 0 \iff \rho c(x) = 0
\]

\[
\underline{\mathrm{Pic}}^{\tau}_{Y} \simeq \hat A_1 \times \hat B^{\Gamma}
\]

\[
\begin{aligned}
H^1(Y, \mathcal{O}_Y) &\simeq H^1(A_1, \mathcal{O}_{A_1}) \times H^1(B, \mathcal{O}_B) \simeq t_{\hat A_1} \times t_{\hat B_\Gamma} \\
H^1(X, \mathcal{O}_X) &\simeq H^1(A, \mathcal{O}_A) \times H^1(B, \mathcal{O}_B)
\end{aligned}
\]

\[
H^1(Y, \underline{\mathcal{T}}_Y) = H^1(Y, \mathcal{O}_Y) \otimes (t_A \times t_B) \longrightarrow H^1(Y, \mathcal{O}_Y) \otimes t_{A_1} \longleftarrow t_{\hat A_1} \otimes t_{A_1}
\]

\[
\downarrow
\]

\[
H^1(X, \underline{\mathcal{T}}_X) = H^1(X, \mathcal{O}_X) \otimes (t_A \times t_B)
\]

\[
(t_{\hat A} \times t_{\hat B}) \otimes (t_A \times t_B) ,
\qquad t_{\hat B} \otimes t_A , \quad t_{\hat B} \otimes t_B
\]

\[
\hat A \leftarrow \hat A_1 , \qquad A \to A_1 , \qquad \hat A \leftarrow \hat A_1
\]

\[
t_A \otimes t_{\hat A} \rightleftarrows t_{A_1} \otimes t_{\hat A_1}
\]

\[
0 \to \mu_2 \to \underline{\mathrm{Pic}}^{*}_{Y} \to \underline{\mathrm{Pic}}^{\Gamma}_{X} \to \mu_2
\]

\[
\text{\struck{$Y = \operatorname{Spec} k[t]$ -- origine}} = \operatorname{Spec} k[t, t^{-1}]
\]

\[
(X_1 \amalg_Y X_2)/\mathbb{Z}_2
\]

\[
\uparrow
\]

\[
X_1 \amalg X_2/\mathbb{Z}/2 \simeq X_1
\]

\[ 0 \to \underline{\mathcal{O}}_X^{*} \to \underline{R}^{*}_{X/S} \to \underline{\mathcal{D}}_{X/S} \to 0 \]

\[ \varepsilon_*\bigl(f_*(\underline{\mathcal{O}}_Y^{*}) / \underline{\mathcal{O}}_S^{*}\bigr) = q_*(\underline{\mathcal{O}}_Y^{*}) / \varepsilon_*(\underline{\mathcal{O}}_S^{*}) \]

\[ 0 \to \varepsilon_*\bigl(f_*(\underline{\mathcal{O}}_Y^{*}) / \underline{\mathcal{O}}_S^{*}\bigr) \to \underline{\mathcal{D}}_{X'/S} \to p_*(\underline{\mathcal{D}}_{X/S}) \to 0 \]

\[ 0 \to f_*(\underline{\mathcal{O}}_Y^{*}) / \underline{\mathcal{O}}_S^{*} \to f'_*(\underline{\mathcal{D}}_{X'/S}) \to f_*(\underline{\mathcal{D}}_{X/S}) \to 0 \]

\[ 0 \to f_*(\underline{\mathcal{O}}_Y^{*}) / \underline{\mathcal{O}}_S^{*} \to \underline{\mathrm{Div}}(X'/S) \to \underline{\mathrm{Div}}(X/S) \to 0 \]

\[ 0 \to \underline{\mathcal{O}}_{X'}^{*} \to p_*(\underline{\mathcal{O}}_X^{*}) \to \varepsilon_*\bigl(f_*(\underline{\mathcal{O}}_Y^{*}) / \underline{\mathcal{O}}_S^{*}\bigr) \]

\[ x \cdot D - D \ \text{lin. équivalent à } 0 \]

\[ \underline{\mathrm{NS}}^{0}_{X/S} = \underline{\mathrm{Pic}}_{X/S} / \underline{\mathrm{Pic}}^{\tau}_{X/S} \]

\[ \underline{\mathrm{Pic}}^{\tau}_{X/k} \to \underline{\mathrm{Pic}}^{\tau}_{Y/k} \ \text{injectif ? \uncertain{Dom} ?} \]

\[ \underline{\mathrm{Pic}}_{X/k} \to \underline{\mathrm{Pic}}^{\tau}_{Y/k} \ \text{\uncertain{surjectif}} \]

\[ \underline{\mathrm{Pic}}^{\tau}_{X/S} \ \text{de type fini} \Longleftrightarrow \underline{\mathrm{Pic}}^{\tau}_{X'/S'} \]