\[
\underline{V}_{C/S} = f^*\bigl(s^*(\underline{V}_{C/S})\bigr) = f^*(\mathcal{L}_{C/S})
\qquad [\,\mathcal{L} = \text{faisceau de Lie}\,]
\]

\[
s^*(\underline{L}) \simeq s^*(\underline{L}\otimes_{C}\underline{O}_{S'})
\simeq s^*(\underline{N}_{C/S'}) \simeq s^*(\underline{V}_{C/S}) = \mathcal{L}
\]

\[
s^*(\underline{L}^{\otimes n}) \simeq \mathcal{L}_{C/S}^{\otimes n}.
\]

\[
\sigma\colon \underline{O}_C \longrightarrow \underline{L}, \qquad
\sigma\in\Gamma(C,\underline{L})
\]

\[
\cdots \longrightarrow \underline{L}^{\otimes n} \longrightarrow
\underline{L}^{\otimes(n+1)} \longrightarrow \cdots \longrightarrow
\underline{L}^{\otimes(n+k)} \longrightarrow \cdots
\]

\[
\cdots \longrightarrow f_*(\underline{L}^{\otimes n}) \longrightarrow
f_*(\underline{L}^{\otimes(n+1)}) \longrightarrow \cdots
\]

\[
\cdots \longrightarrow \mathcal{E}_n \longrightarrow \mathcal{E}_{n+1} \longrightarrow \cdots
\]

\[
\mathcal{E}_n = f_*(\underline{L}^{\otimes n}).
\]

\[
\mathcal{E}_0 = \mathcal{E}_1 \subset \mathcal{E}_2 \subset \mathcal{E}_3 \subset \cdots
\]

\[
\mathcal{E}_0 = \underline{O}_S .
\]

\[
\mathcal{L}_n \simeq \mathcal{E}_{n}/\mathcal{E}_{n-1}
\]

\[
\mathcal{L}_n\otimes\mathcal{L}_m \simeq \mathcal{L}_{m+n} \quad \text{si } n,m
\underset{\neq 1}{\geqslant 0}
\]

\[
\mathcal{E} = \varinjlim \mathcal{E}_n = f_*(\varinjlim \underline{L}^{\otimes n})
\]

\[
\mathcal{E}_m\times\mathcal{E}_n \longrightarrow \mathcal{E}_{m+n}
\]

\[
\underline{\mathcal{L}}_n\otimes\underline{\mathcal{L}}_m \longrightarrow
\underline{\mathcal{L}}_{n+m}, \qquad n,m \underset{\neq 1}{\geqslant 0},
\]

\[
\underline{\mathcal{L}}\otimes\underline{\mathcal{L}}_n \longrightarrow
\underline{\mathcal{L}}_{n+1}, \qquad n\geqslant 2 .
\]

\[
\underline{\mathcal{L}}\otimes\underline{\mathcal{L}}_n \simeq
\underline{\mathcal{L}}_{n+1} \quad \text{pour } n\geqslant 2, \text{ donc}
\]

\[
\mathcal{L}_n \simeq \mathcal{L}^{n-2}\otimes\mathcal{L}_2, \qquad \text{donc pour }
n\geqslant 2,
\]

\[
\mathcal{L}_2 \simeq \mathcal{L}^{\otimes 2} .
\]

\[
\cdots\longrightarrow\underline{L}^{\otimes n}\longrightarrow
\underline{L}^{\otimes(n+1)}\longrightarrow
(\underline{V}_{C/S})^{\otimes(n+1)}\otimes_{\underline{O}_C}\underline{O}_{S'}
\longrightarrow 0
\]

\[
0\longrightarrow\mathcal{E}_n\longrightarrow\mathcal{E}_{n+1}\longrightarrow
\mathcal{L}^{\otimes(n+1)}\longrightarrow R^1f_*(\underline{L}^{\otimes n})
\]

\[
\boxed{\ \mathcal{L}_{n+1}\simeq\mathcal{L}^{\otimes(n+1)} \qquad n\geqslant 2\ }
\]

\[
C \longrightarrow \mathbf{P}(\mathcal{E}_3)
\]

\[
\mathbf{P}_3(\mathbf{P}(\mathcal{E}_3)) \simeq
\mathbf{P}(\operatorname{Symm}_3(\check{\mathcal{E}}_3)) .
\]

\[
a x^{?} + b y^3 + c xy + d y^2 + e x + f y + g = 0
\]

\[
1,\ y,\ x,\ y^2,\ xy,\ y^3
\]

\[
\varphi(x,y) = \boxed{\,x^2 + a y^3 + b xy + c y^2 + d x + e y + f = 0\,}
\]

\[
\Phi(x,y,z) = x^2 z + a y^3 + b xyz + c y^2 z + d x z^2 + f z^3 = 0
\]

\[
\delta(a,b,c,d,e,f)
\]

\[
\mathfrak{M} = \operatorname{Spec}\bigl(\mathbf{Z}[A,B,C,D,E,F][\delta^{-1}]\bigr)
\]

\[
\begin{pmatrix} 1 & \alpha & \gamma \\ 0 & \lambda & \beta \\ 0 & 0 & \mu \end{pmatrix}
\]

\[
X^2 + AY^3 + BXY^{2} + CY^2 + DX + EY + F
\]

\[
Y' = \lambda Y + \alpha, \qquad X' = \mu X + \beta Y + \gamma
\]

\[
\left(\lambda,\ \begin{pmatrix} 1 & \alpha & \beta \\ 0 & \lambda^2 & \gamma \\ 0 & 0 & \lambda^3 \end{pmatrix}\right)
\qquad
\begin{array}{l}
\lambda\in\Gamma(\mathbf{G}_{m,S}/S)=\Gamma(S,\underline{O}_S^*)\\
\alpha,\beta,\gamma\in\Gamma(S,\underline{O}_S)
\end{array}
\]

\[
\begin{pmatrix} 1 & \alpha & \beta \\ 0 & \lambda & \gamma \\ 0 & 0 & \mu \end{pmatrix}
\quad\text{pour}\quad
\lambda = \left(\frac{\mu}{\lambda}\right)^2, \quad
\mu = \left(\frac{\mu}{\lambda}\right)^3 \qquad\text{i.e.}\quad
\boxed{\mu^2 - \lambda^3 = 0}
\]

\[
x^2 - y^3 + bxy + cy^2 + dx + ey + f = 0
\]

\[
x^2 = y^3 + axy + by^2 + cx + dy + e
\]

\[
x^2 = y^3 - axy + by^2 - cx + dy + e
\]

\[
\boxed{\,x^2 = y^3 + ay^2 + by + c\,}
\]

\[
\begin{pmatrix} 1 & \alpha & 0 \\ 0 & \lambda^2 & 0 \\ 0 & 0 & \lambda^3 \end{pmatrix}
\quad\text{i.e.}\quad
\begin{pmatrix} 1 & \alpha & 0 \\ 0 & u & 0 \\ 0 & 0 & v \end{pmatrix}
\quad\text{avec}\quad u^3 - v^2 = 0 .
\]

\[
j\equiv 1 \quad \Bigl(\frac{t}{t-1}\Bigr)^3 \in O^6O^* \qquad
\Bigl(\frac{t-1}{t}\Bigr)^2 \in O^6O^* \qquad
\sqrt[6]{\frac{t}{t-1}}
\]

\[
\sqrt[2]{\frac{t}{t-1}} \qquad \sqrt[3]{\frac{t-1}{t}} \qquad \tfrac12 \qquad \tfrac13
\]

\[
\lambda^{3/6}\cdot\lambda^{-2/6} = \lambda^{1/6} \qquad
s^6 = \frac{t}{t-1}
\]

\[
\frac{-27c^2}{b^3-27c^2} = j \qquad 27c^2(j-1) = b^3 j \qquad
\frac{b^3}{c^2} = j
\]

\[
\begin{array}{ccc} L & L^2 & L^3 \\ & b & c \end{array}
\qquad
\boxed{\ \frac{b^3}{c^2} = 27\,\frac{j-1}{j}\ }
\]

\[
b = 27\,\frac{j-1}{j} \qquad c = 27\,\frac{j-1}{j}
\]

\[
\boxed{\ x^3 + 27\,\frac{j-1}{j}\,(x+1) = 0\ }
\]

\[
\underline{\mathrm{Hom}}_{\underline{A}}(E,\underline{A})\times
\underline{\mathrm{Hom}}_{\underline{O}}(\underline{A},\underline{O})
\]

\[
\prod_{i\neq j}(x_i-x_j) = \prod_i \Bigl(\frac{f(x)}{x-x_i}\Bigr)_{x\to x_i}
= \prod_i f'(x_i)
\]

\[
f(x_i) =
\]

\[
3x^2 + 2ax + b
\]

\[
(3x_1^2+2ax_1+b)(3x_2^2+2ax_2+b)(3x_3^2+2ax_3+b)
\]

\[
j = -\frac{27c^2}{\Delta}
\]

\[
\Delta = \prod(3x_i^2+b) = 27(x_1x_2x_3)^2 + 9b\bigl(\textstyle\sum(x_1x_2)^2
+ 3b^2\sum x_i^2\bigr) + b^3
\]

\[
(x_1x_2x_3)^2 = c^2
\]

\[
\textstyle\sum(x_1x_2)^2 = \bigl(\sum x_1x_2\bigr)^2 - 2\sum x_1^2x_2x_3
\]

\[
\textstyle\sum x_i^2x_2x_3 = \bigl(\sum x_i^2\bigr)\bigl(\sum x_2x_3
\]

\[
\boxed{\ \Delta = b^3 - 27c^2\ }\ ?
\qquad x_1^2x_2^2 + \qquad \textstyle\prod 3x_i^2+b
\]

\[
3\textstyle\sum(x_1x_2)^2 + \bigl(\sum x_1x_2\bigr)\bigl(\sum x_i^2\bigr)
\]