\[
  0 \to \mu_2 \to U \xrightarrow{\;2\;} U \to 0 ,
\]

\[
  \mathbb{D}_S = \mathcal{O}(2)_S \quad \text{\uncertain{au} 4-groupe} \quad
  (\mathbb{Z}/2)_S \tfrac{1}{2}\, \mu_{2,S} \simeq (\mathbb{Z}/2)_S \times (\mu_2)_S
\]

\[
  \Updownarrow
\]

\[
  \Downarrow
\]

\[
  X \longleftrightarrow M \longleftrightarrow (X \subset C), \qquad C = \check{\mathbb{P}}(\gamma M)
\]

\[
  \tilde{\sigma}(u) = \sigma u^{*} \sigma = (\sigma u \sigma)^{*} = u - \mathrm{Tr}(u \sigma)\, \sigma
\]

\[
  (1) \qquad
  \begin{aligned}
    \sigma(x,y) &= (y, x), \qquad \sigma'(x,y) = (\zeta y, \zeta^{-1} x), \\
    u(x,y) &= (\zeta x, \zeta^{-1} y) .
  \end{aligned}
\]

\[
  (2) \qquad
  \begin{aligned}
    \mathrm{Sym}^2(\check{E}) \simeq{}& \check{D}_1^{\otimes 2} \oplus \check{D}_2^{\otimes 2} \oplus \check{D}_0^{\otimes 2} \\
    &\oplus \check{D}_1 \otimes \check{D}_2 \oplus \check{D}_1 \otimes \check{D}_0 + \check{D}_2 \otimes \check{D}_0
  \end{aligned}
\]

\[
  (3) \qquad \zeta^{2},\ \zeta^{-2},\ 1,\ 1,\ \zeta^{-1},\ \uncertain{\zeta}
\]

\[
  (4) \qquad \zeta^{4} = 1 \quad \text{i.e.} \quad \zeta^{2} = -1, \quad \text{ou encore } \alpha = 0
\]

\[
  (\zeta^{2} - 1)(\zeta^{2} + 1) = 0
\]

\[
  (1,1),\ (-1,1),\ (-1,-1),\ (1,-1)
\]

\[
  S' = \mathrm{Spec}\, \mathcal{O}_S[T]/(T^2 - \alpha T + 1)\ \ldots)
\]

\[
  (5) \qquad Q(x,y,z) = a z^2 + b xy
\]

\[
  \begin{pmatrix} 0 & b & 0 \\ b & 0 & 0 \\ 0 & 0 & 2a \end{pmatrix}
\]

\[
  (6) \qquad \delta(Q) = 2ab^2
\]

\[
  (7) \qquad \delta'(Q) = ab^2
\]

\[
  (8) \qquad a + b = 0
\]

\[
  (9) \qquad z^2 - xy = 0
\]

\[
  xy - 1
\]

\[
  \alpha(\alpha^2 - 4)(\alpha + 1) \quad \text{inversible}
\]

\[
  Q(x,y) = a x^2 + 2b\, xy + c y^2 + u x + v y + \lambda
\]

\[
  \begin{pmatrix}
    1 & 1 & 1 & 1 & 1 & 1 \\
    \zeta^{2} & 1 & \zeta^{-2} & \zeta & \zeta^{-1} & 1 \\
    \zeta^{-2} & 1 & \zeta^{2} & \zeta^{-1} & \zeta & 1 \\
    \zeta^{4} & 1 & \zeta^{-4} & \zeta^{2} & \zeta^{-2} & 1 \\
    \zeta^{-4} & 1 & \zeta^{4} & \zeta^{-2} & \zeta^{2} & 1
  \end{pmatrix}
\]

\[
  \det
  \begin{pmatrix}
    1 & 1 & 1 & 1 & 1 \\
    \zeta^{2} & \zeta^{-2} & \zeta & \zeta^{-1} & 1 \\
    \zeta^{-2} & \zeta^{2} & \zeta^{-1} & \zeta & 1 \\
    \zeta^{4} & \zeta^{-4} & \zeta^{2} & \zeta^{-2} & 1 \\
    \zeta^{-4} & \zeta^{4} & \zeta^{-2} & \zeta^{2} & 1
  \end{pmatrix}
  = \Delta(\zeta), \qquad \text{où } \Delta \in \mathbb{Z}[T, T^{-1}]
\]

\[
  \frac{(\zeta^{4} - 1)(\zeta^{3} - 1)}{\zeta - 1} \in \mathbb{Z}[\zeta]
\]

\[
  \frac{(\zeta^{2}-1)(\zeta^{3}-1)}{\zeta - 1} = (\zeta^{2}+1)(\zeta^{2}-1)(1 + \zeta + \zeta^{2}), \quad \text{et} \equiv \text{par}
\]

\[
  (\zeta^{2}+1)(\zeta^{2}-1)^{2}(1 + \zeta + \zeta^{2}) = \alpha(\alpha^{2} - 4)(\alpha + 1)
\]

\[
  (3) \qquad \text{\struck{$q(x,y)$}}\ \ q_0(x,y) = x^2 + y^2 - \alpha xy
\]

\[
  (4) \qquad \delta_0 = \alpha^2 - 4 = (\alpha - 2)(\alpha + 2)
\]

\[
  (2\,\text{bis}) \qquad q_0(x,y,z) = x^2 + y^2 \ \text{\struck{$+ \ill{}$}} + 0 z^2 - yz - \text{\struck{\ill{}}}\, xz - \alpha xy
\]

\[
  (5) \qquad
  \begin{pmatrix} 2 & -\alpha & -1 \\ -\alpha & 2 & -1 \\ -1 & -1 & 0 \end{pmatrix}
\]

\[
  (6) \qquad \delta = \text{\struck{\ill{}}} -2\alpha - 2 - 2 = -2(\alpha + 2)
\]

\[
  (6') \qquad \delta' = -(\alpha + 2)
\]

\[
  (7) \qquad \zeta^2 - \alpha\zeta + 1 = 0 \quad \text{i.e.} \quad \zeta + \zeta^{-1} = \alpha
\]

\[
  (8) \qquad q_0(x,y) = (x - \zeta y)(x - \zeta^{-1} y)
\]

\[
  (9) \qquad q(x,y,z) = (x - \zeta y)(x - \zeta^{-1} y) - (x + y) z
\]

\[
  \left\{
  \begin{aligned}
    f_1 &= (1, \zeta, 0) = e_1 + \zeta e_2 \\
    f_2 &= (1, 0, 1 - \zeta) = e_1 + (1 - \zeta) e_3 \\
    f_3 &= (0, 0, 1) = e_3
  \end{aligned}
  \right.
\]

\[
  \begin{aligned}
    e_1 &= f_2 - (1 - \zeta) f_3 = f_2 + (\zeta - 1) f_3 \\
    e_2 &= \zeta^{-1}(f_1 - e_1) = \zeta^{-1} f_1 - \zeta^{-1} f_2 + (1 - \zeta^{-1}) f_3 \\
    e_3 &= f_3
  \end{aligned}
\]

\[
  (1) \qquad
  \left\{
  \begin{aligned}
    s_0 &= (0,0) \\
    s_1 &= (1,0) \\
    s_{-1} &= (0,1) \\
    s_2 &= (1 + \alpha, 1) \\
    s_{-2} &= (1, 1 + \alpha)
  \end{aligned}
  \right.
\]

\[
  q(x,y) = a x^2 + b xy + c y^2 + u x + v y + d
\]

\[
  \left.
  \begin{aligned}
    q(s_0) &= d = 0 \\
    q(s_1) &= a + u = 0 && \text{i.e. } u = -a \\
    q(s_{-1}) &= c + v = 0 && \phantom{\text{i.e. }} v = -b
  \end{aligned}
  \right\}
  \quad q(x,y) = a(x^2 - x) + c(y^2 - y) + bxy
\]

\[
  \left\{
  \begin{aligned}
    q(s_2) &= a(1+\alpha)\alpha + b(1+\alpha) = (1+\alpha)(\alpha a + b) = 0 \\
    q(s_{-2}) &= c(1+\alpha)\alpha + b(1+\alpha) = (1+\alpha)(\alpha c + b) = 0
  \end{aligned}
  \right.
\]

\[
  b = -\alpha a \ \text{\struck{\ill{}}} = -\alpha c
\]

\[
  a = c, \quad b = -\alpha a,
\]

\[
  (2) \qquad q_0(x,y) = (x^2 - x) + (y^2 - y) - \alpha xy
\]

\[
  Ax + By + Cz = 0 \qquad \text{\struck{$2x - \alpha y =$}}
\]

\[
  \begin{aligned}
    A &= 2 - \alpha\zeta = 1 - \zeta^2 \ \text{\struck{$= \ill{}$}} = (1 - \zeta)(1 + \zeta), \\
    B &= -\alpha + 2\zeta = \zeta - \zeta^{-1} = \zeta^{-1}(1 - \zeta^2) = -\zeta^{-1}(1 - \zeta)(1 + \zeta), \\
    C &= -(1 + \zeta)
  \end{aligned}
\]

\[
  H_\zeta : \quad
  \left\{
  \begin{aligned}
    &(1 - \zeta)x - \zeta^{-1}(1 - \zeta) y - z = 0 \quad \text{i.e.} \\
    &z = (1 - \zeta)(x - \zeta^{-1} y)
  \end{aligned}
  \right.
\]

\[
  D_\zeta \subset H_\zeta \subset E, \qquad H_\zeta \simeq V
\]

\[
  (E/H_\zeta)^{\otimes -1} \otimes H_\zeta/D_\zeta, \qquad
  E/H_\zeta \simeq \mathcal{O}_S\, e_3, \quad H_\zeta/D_\zeta \simeq \mathcal{O}_S\, e_1
\]

\[
(1)\qquad \Omega = \mathrm{Spec}(B)
\]

\[
(2)\qquad \check{L} = B/\mathcal{O}_S \qquad 0 \to \mathcal{O}_S \xrightarrow{\;i\;} B \to \check{L} \to 0
\]

\[
(3)\qquad 0 \to L \to \check{B} \xrightarrow{\;\pi\;} \mathcal{O}_S \to 0
\]

\[
(4)\qquad E = \check{\mathbb{V}}(\check{B}) \simeq \mathbb{V}(B) = \mathrm{Spec}\bigl(\underline{\mathrm{Sym}}_{\mathcal{O}_S}(B)\bigr)
= \mathrm{Spec}\bigl(\underline{\mathrm{Sym}}_{\mathcal{O}_S} B/(e_B - 1)\bigr)
\]

\[
(5)\qquad \Omega(S) \hookrightarrow E(S)
\]

\[
(6)\qquad \Omega \hookrightarrow E
\]

\[
\begin{aligned}
&(7) && 0 \to V \to \check{C} \to \mathcal{O}_S \to 0\\
&\text{\struck{(8)}} && 0 \to \mathcal{O}_S \to \check{C} \to \check{V} \to 0
\end{aligned}
\]

\[
F = \mathrm{Spec}\bigl(\underline{\mathrm{Sym}}\, \check{C}/(e_C - 1)\bigr)
\]

\[
\Omega \longrightarrow F
\]

\[
\underline{\mathrm{Sym}}(C)/(e_C - 1) \longrightarrow B
\]

\[
C \xrightarrow{\;\varphi\;} B
\]

\[
\varphi(e_C) = e_B
\]

\[
(9)\qquad \mathrm{Hom}_S(\Omega, F) \simeq \mathrm{Homaff}_S(E, F)
\]

\[
\text{\struck{\ill{}}}\ (10)\qquad \underline{\mathrm{Hom}}_S(\Omega, F) \simeq \underline{\mathrm{Homaff}}_S(E, F) .
\]

\[
(11)\qquad B \simeq \mathcal{O}_S \oplus \check{L} ,
\]

\[
\check{L} \otimes \check{L} \longrightarrow B \simeq \mathcal{O}_S + \check{L}
\]

\[
(12)\qquad
\begin{cases}
b \in \Gamma\,\underline{\mathrm{Hom}}(L^{\otimes(-2)}, L^{-1}) \simeq \Gamma L\\
c \in \Gamma\,\underline{\mathrm{Hom}}(L^{\otimes(-2)}, \mathcal{O}_S) \simeq \Gamma L^{\otimes 2} .
\end{cases}
\]

\[
(13)\qquad
\begin{cases}
(\lambda \oplus s(u))(\mu \oplus s(v)) = (\lambda\mu - cuv) \oplus s(\lambda v + \mu u + buv) .\\
\text{i.e. } s(u)s(v) = -cuv \oplus s(buv)
\end{cases}
\]

\[
\mathrm{tr}(\lambda + u) = 2\lambda + \mathrm{tr}\, u
\]

\[
\mu \oplus v \longmapsto -cuv \oplus s(\mu u + buv)
\]

\[
\mu_u = \begin{pmatrix} 0 & -cu\\ \uncertain{u} & -bu \end{pmatrix} ,
\qquad
\mu_{\lambda \oplus u} = \begin{pmatrix} \lambda & -cu\\ u & \lambda + bu \end{pmatrix}
\]

\[
(14)\qquad
\begin{cases}
\mathrm{tr}\, u = bu\\
N u = cu^2
\end{cases}
\]

\[
(15)\qquad
\begin{cases}
\mathrm{tr}(\lambda \oplus u) = 2\lambda + bu\\
N(\lambda \oplus u) = \lambda^2 + \lambda bu + cu^2
\end{cases}
\]

\[
\boxed{x^2 - bx + c = 0}
\]

\[
(16)\qquad s' = s + \tau \qquad \tau \in \Gamma(E) \qquad \text{i.e. } s'(u) = s(u) + \tau u \ \text{pour } u \in \Gamma\check{L}
\]

\[
b' = b(s'), \qquad c' = c(s')
\]

\[
\begin{aligned}
s'(u)s'(v) &= -c'uv + s'(b'uv) = -c'uv + s(b'uv) + \tau b'uv\\
&= (\tau b' - c')uv \oplus s(b'uv)
\end{aligned}
\]

\[
\begin{aligned}
\Vert\qquad&\\
(s(u) \oplus \tau u)(s(v) \oplus \tau v) &= s(u)s(v) + (\tau u)s v + \tau(v) s u + \text{\struck{$(\tau u)(\tau v)$}}\\
&= \text{\struck{\ill{}}}\\
&= (\tau^2 - c)uv \oplus s\bigl((b + 2\tau\ \text{\struck{\ill{}}})uv\bigr)
\end{aligned}
\]

\[
\begin{cases}
b' = b + 2\tau\\
\tau b' - c' = \tau^2 - c
\end{cases}
\qquad
\begin{aligned}
\text{i.e. } c' &= c + \tau b' - \tau^2\\
&= c + \tau b + 2\tau^2 - \tau^2\\
&= c + \tau b + \tau^2
\end{aligned}
\]

\[
(17)\qquad
\begin{cases}
b' = b + 2\tau\\
c' = c + \tau b + \tau^2 .
\end{cases}
\]

\[
(18)\qquad \mathfrak{D} = b^2 - 4c \in \Gamma(S, L^{\otimes 2}) ,
\]

\[
\begin{aligned}
\mathfrak{D}(s') &= (b + 2\tau)^2 - 4(c + \tau b + \tau^2)\\
&= b^2 + (4\tau^2 + 4\tau b) - 4(c + \tau b + \tau^2)\\
&= b^2 - 4c = \mathfrak{D}(s)
\end{aligned}
\]

\[
\text{\struck{$N(\xi + \eta) = N(\xi) + N(\eta) + \mathrm{tr}(\xi\eta)$}}
\]

\[
\begin{aligned}
&\text{\struck{$N(\xi + \eta) = (\lambda + \mu)^2 + b(\lambda + \mu)(u + v) + c(u + v)^2$}}\\
&\text{\struck{$\quad = (\lambda^2 + b\lambda u + cu^2) + (\mu^2 + b\mu v + cv^2) + 2cuv + 2\lambda\mu + b(\lambda v + \mu u)$}}\\
&\text{\struck{$\quad = N(\xi) + N(\eta) + 2cuv + 2\lambda\mu + b(\lambda v + \mu u)$}}
\end{aligned}
\]

\[
(19)\qquad \Delta(N_{\Omega/S}) = -\text{\struck{$4$}}\,\mathfrak{D}(\Omega/S)
\]

\[
\begin{aligned}
&(20) && \varphi_{\Omega/S}(\xi, \eta) = \mathrm{tr}\,\xi\, \mathrm{tr}\,\eta - \mathrm{tr}(\xi\eta)\\
&\text{[ou variante (21)} && \varphi_{\Omega/S}(\xi, 1) = \mathrm{tr}\,\xi\ ]
\end{aligned}
\]

\[
(22)\qquad N(\xi + \eta) = N(\xi) + N(\eta) + \bigl(\mathrm{tr}\,\xi\,\mathrm{tr}\,\eta - \mathrm{tr}(\xi\eta)\bigr) .
\]

\[
\wedge^2\varphi(\xi \wedge \xi', \eta \wedge \eta') = \det\begin{pmatrix} \varphi(\xi, \eta) & \varphi(\xi, \eta')\\ \varphi(\xi', \eta) & \varphi(\xi', \eta') \end{pmatrix}
\]

\[
\wedge^2\varphi(e_B \wedge \xi, e_B \wedge \xi) = \det\begin{pmatrix} (\mathrm{tr}\,\xi)^2 - \mathrm{tr}(\xi^2) & \mathrm{tr}\,\xi\\ \mathrm{tr}\,\xi & 2 \end{pmatrix} = \text{\struck{$2$}}(\mathrm{tr}\,\xi)^2 - 2\,\mathrm{tr}(\xi^2)
\]

\[
\text{\struck{$= (\mathrm{tr}\,\xi)^2 \ldots - (\mathrm{tr}\,\eta)^2$}}
\]

\[
\xi^2 = s(u)^2 = -cu^2 \oplus s(bu^2), \qquad \mathrm{tr}\,\xi^2 = -2cu^2 + b^2u^2
\]

\[
\begin{aligned}
\wedge^2\varphi(1 \wedge s(u), 1 \wedge s(u)) &= \text{\struck{\ill{}}}\ (bu)^2 - 2(-2cu^2 + b^2u^2)\\
&= (4c - b^2)u^2 = -\mathfrak{D}u^2 , \quad \text{cqfd.}
\end{aligned}
\]

\[
(23)\qquad L \xrightarrow[\sim]{\;\alpha\;} \mathcal{O}_S
\]

\[
E_\Omega \xrightarrow{\;\sim\;} \mathbb{E}^1_S ,
\]

\[
(24)\qquad t^2 - bt + c = 0
\]

\[
(25)\qquad b = \mathrm{tr}\, t, \qquad c = N(t) .
\]

\[
L^{\otimes 2} \xrightarrow[\sim]{\;s^2\;} \mathcal{O}_S
\]

\[
(26)\qquad \mathfrak{D}(\Omega/S) = (b^2 - 4c)s^2 \qquad \text{identifié, par } s, \text{ à } b^2 - 4c .
\]

\[
(27)\qquad E \xleftarrow[\sim]{\;\alpha_s\;} \check{\mathbb{V}}L
\]

\[
\Sigma^n(\alpha_s) : \Sigma^n \check{\mathbb{V}}(L) = \check{\mathbb{V}}\Bigl(\sum_{1 \le i \le n} L^{\otimes i}\Bigr) \xrightarrow{\;\sim\;} \Sigma^n E
\]

\[
(28)\qquad \Sigma^n \check{\mathbb{V}}L \simeq \check{\mathbb{V}}L^n/\mathfrak{S}_n \longrightarrow \check{\mathbb{V}}\Bigl(\sum_{1 \le i \le n} L^{\otimes i}\Bigr)
\]

\[
(29)\qquad \sigma_n : \check{\mathbb{V}}(L^n) \longrightarrow \check{\mathbb{V}}\Bigl(\sum_{1 \le i \le n} L^{\otimes i}\Bigr)
\]

\[
(30)\qquad \sigma_n = (\sigma_i)_{1 \le i \le n} \qquad \sigma_i(x_1, \ldots, x_n) = \sum x_1 x_2 \cdots x_i
\]

\[
s' = s + \tau \qquad (\tau \in \Gamma L)
\]

\[
\alpha_{s'} = \alpha_s + \mathrm{trans}(\tau)
\]

\[
(31)\qquad \Sigma^n(\alpha_{s'}) = \Sigma^n(\alpha_s)\,\underbrace{\Sigma^n(\mathrm{trans}\,\tau)}_{\varphi_n(\tau)}
\]

\[
\begin{aligned}
\varphi_n(\tau) : \Sigma^n(\check{\mathbb{V}}L) &\xrightarrow{\;\sim\;} \Sigma^n \check{\mathbb{V}}L\\
\Vert\qquad&\qquad\qquad \Vert\\
\textstyle\sum L^{\otimes i} &\longrightarrow \textstyle\sum L^{\otimes i}
\end{aligned}
\]

\[
\Bigl(\sum (x_1 + \tau)(x_2 + \tau) \cdots (x_i + \tau)\Bigr)_{1 \le i \le n} = \varphi_n(\tau)\bigl(\underbrace{\textstyle\sum x_1 \cdots x_i}_{\sigma_i(x)}\bigr)_i
\]

\[
\Vert
\]

\[
\text{\struck{$\sigma_1(x) + n\tau,\ \sigma_2(x) + (n-1)\tau\sigma_1(x) + \tfrac{n(n-1)}{2}\tau^2,\ \sigma_3(x) + \ldots$}}
\]

\[
\begin{aligned}
\Bigl(\prod_{1 \le i \le n} (1 + X\tau + X_i)\Bigr) - 1 &= \Bigl(\sum_{0 \le p \le n} (1 + \tau)^{n-p}\sigma_p\Bigr) - 1\\
&= \text{\struck{\ill{}}} \sum_{\substack{0 \le p, q \le n\\ p + q > 0}} \frac{(n-p)!}{q!\,(n-p-q)!}\,\tau^q\sigma_p
\end{aligned}
\]

\[
(32)\qquad \varphi_n(\tau)(\sigma_1, \ldots, \sigma_n) = \bigl(\varphi_n(\tau)_i(\sigma_1, \ldots, \sigma_n)\bigr)_{1 \le i \le n}
\]

\[
(33)\qquad \varphi_n(\tau)_i(\sigma_1, \ldots, \sigma_n) = \sum_{\substack{p, q \ge 0\\ p + q = i}} \frac{(n-p)!}{(n-i)!\,q!}\,\tau^q\sigma_p \ \text{\struck{$= \ldots$}}
\]

\[
\check{\mathbb{V}}(L) \xrightarrow{\;\varphi_n\;} \mathrm{Aff}\Bigl(\check{\mathbb{V}}\bigl(\textstyle\sum_{1 \le i \le n} L^{\otimes i}\bigr)\Bigr)
\]

\[
\begin{cases}
s \longmapsto \Sigma^n(\alpha_s)\\
E \longrightarrow \mathrm{Isom}\Bigl(\check{\mathbb{V}}\bigl(\sum_{1 \le i \le n} L^{\otimes i}\bigr), \Sigma^n(E)\Bigr)
\end{cases}
\]

\[
(34)\qquad E \wedge^{\check{\mathbb{V}}(L)} \check{\mathbb{V}}\Bigl(\sum_{1 \le i \le n} L^{\otimes i}\Bigr) \simeq \Sigma^n(E)
\]

\[
(35)\qquad \varphi_2(\tau)(\sigma_1, \sigma_2) = (\sigma_1 + 2\tau,\ \sigma_2 + \tau\sigma_1 + \tau^2)
\]

\[
(36)\qquad \varphi_2^0(\tau)(\sigma_1, \sigma_2) = (\sigma_1,\ \sigma_2 + \tau\sigma_1)
\]

\[
\psi : \check{\mathbb{V}}(L) \longrightarrow \underline{\mathrm{Autom}}\ \text{d'ex}\bigl(0 \to L^{\otimes 2} \to L \otimes \check{B} \to L \to 0\bigr)
\]

\[
(37)\qquad 0 \to L \to \check{B} \to \mathcal{O}_S \to 0
\]

\[
\check{B} \simeq \text{\struck{$L$}}\ \mathcal{O}_S \oplus L
\]

\[
(38)\qquad \tilde{\alpha}_s : \check{B} \otimes L \simeq L \oplus L^{\otimes 2}
\]

\[
\text{Donc } \Sigma^2(F) = F \wedge^{\check{\mathbb{V}}(L), \varphi_2^0} \check{\mathbb{V}}(L + L^{\otimes 2})
\]

\[
\Sigma^2(F)/\mathbb{V}(L^{\otimes 2})
\]

\[
(39)\qquad \Sigma^2(F)/\mathbb{V}(L^{\otimes 2}) \simeq 2F
\]

\[
(40)\qquad \Sigma^2(F) \xrightarrow{\;\underline{b}\;} 2F
\]

\[
(41)\qquad \underline{b}_{\Omega(s_1, s_2)} = s_1 + s_2 \quad (\in \Gamma(2F)) .
\]

\[
(42)\qquad b(s) = \underline{b}(s) - 2s
\]

\[
(43)\qquad b(s) = s_1 + s_2 - 2s = (s_1 - s) + (s_2 - s)
\]

\[
(44)\qquad \Sigma(\underline{b}) \ \text{\struck{\ill{}}}
\]

\[
c(\underline{b}) = \mathrm{cl}(\Sigma(\underline{b})) \in H^1(S, L^{\otimes 2})
\]

\[
(45)\qquad 0 \to \check{B} \otimes L \longrightarrow V^{(2)} \longrightarrow \mathcal{O}_S \to 0
\]

\[
(46)\qquad \Sigma^2(F) = \text{fibré affine associé à l'extension } V^{(2)} = \mathrm{Sym}^2(\check{B}) \text{ de } \mathcal{O}_S \text{ par } V^{(1)} = \check{B} .
\]

\[
(47)\qquad 0 \to L \xrightarrow{\;i_1\;} V \xrightarrow{\;\pi_1\;} \mathcal{O}_S \to 0 \qquad (V = \check{B})
\]

\[
(48)\qquad F = \pi_1^{-1}(1) \subset \check{\mathbb{V}}(V)
\]

\[
(49)\qquad 0 \to L \otimes \underline{\mathrm{Sym}}^{n-1}(V) \xrightarrow{\;i_n\;} \underline{\mathrm{Sym}}^n(V) \xrightarrow{\;\pi_n\;} \mathcal{O}_S \to 0
\]

\[
(50)\qquad \underbrace{L^{i} \otimes \underline{\mathrm{Sym}}^{n-i}(V)}_{\mathrm{Fil}^i\, \mathrm{Sym}^n(V)} \subset \underline{\mathrm{Sym}}^n(V) \quad (0 \le i \le n)
\]

\[
\text{\struck{(51)\quad $\Sigma^n F = \pi_n^{-1}(1) \subset \check{\mathbb{V}}(\underline{\mathrm{Sym}}^n V)$}}
\]

\[
\text{\struck{(52)\quad $\underbrace{\Sigma^n(F)}_{\text{\uncertain{puissance} symétrique}} \xrightarrow[\sim]{\;\varphi_n\;} \Sigma^n$}}
\]

\[
\text{\struck{(53)\quad $\Psi_n : F^n \longrightarrow \Sigma^n,\quad \Psi_n(x_1, \ldots, x_n) = x_1$}}
\]

\[
(51)\qquad 0 \to \mathcal{O}_S \xrightarrow{\;i'_1\;} \check{V} \xrightarrow{\;\pi'_1\;} \check{L} \to 0 \qquad (\check{V} = B)
\]

\[
(52)\qquad \check{V} \simeq V \otimes (\det V)^{-1}
\]

\[
(53)\qquad V \otimes V \longrightarrow \det V, \qquad x \otimes y \longmapsto x \wedge y ;
\]

\[
(54)\qquad \det V \xrightarrow[\sim]{\;\alpha\;} L
\]

\[
(55)\qquad \alpha(i_1(\lambda) \wedge x) = \lambda\, \pi_1(x) \qquad \lambda \in \Gamma(-, L),\ x \in \Gamma(-, V)
\]

\[
(56)\qquad \check{V} \xleftarrow[\sim]{\;\alpha\;} V \otimes L^{-1}
\]

\[
(57)\qquad
\begin{cases}
(V \otimes L^{-1}) \otimes V \xrightarrow{\;\varphi\;} \mathcal{O}_S\\
\quad \simeq (V \otimes V) \otimes L^{-1}
\end{cases}
\]

\[
(58)\qquad \varphi(i_1(\lambda) \otimes x \otimes \mu) = \lambda\mu\, \pi_1(x)
\qquad \lambda \in \Gamma L,\ x \in \Gamma V,\ \mu \in \Gamma V
\]

\[
(57\,\text{bis})\qquad V \otimes V \xrightarrow{\;\varphi\;} L
\]

\[
(58\,\text{bis})\qquad \varphi(i_1(\lambda) \otimes x) = \lambda\, \pi_1(x) \qquad \lambda \in \Gamma L,\ x \in \Gamma V
\]

\[
(59)\qquad 0 \to \mathcal{O}_S \xrightarrow{\;i'\;} L^{-1} \otimes V \xrightarrow{\;\pi'\;} \check{L} \to 0
\]

\[
M_D \hookrightarrow \underline{\mathrm{Sym}}^n(\check{V}) \longrightarrow \underline{\mathrm{Sym}}^n(\check{V})/\underline{\mathrm{Sym}}^{n-1}(\check{V}) \simeq L^{\otimes -n}
\]

\[
D \longmapsto M_D
\]

\[
\begin{aligned}
(62)\qquad \Sigma^n(F/S) \simeq{} & \text{fibré affine associé à l'extension}\\
& 0 \to \underline{\mathrm{Sym}}^{n-1}(V) \otimes L \to \underline{\mathrm{Sym}}^n(V) \to \mathcal{O}_S \to 0\\
& \bigl(\text{donc le fibré des translations } \to \underline{\mathrm{Sym}}^{n-1}(V) \otimes L\bigr)
\end{aligned}
\]

\[
(63)\qquad q(e_B) = 1
\]

\[
(64)\qquad \Omega = \mathrm{Spec}(B) ,
\]

\[
(65)\qquad q = N_{B/\mathcal{O}_S}
\]

\[
(66)\qquad B \xrightarrow{\;g\;} C_n
\]

\[
(67)\qquad
\begin{cases}
\text{a) } q(e_B) = 1_{C_n} \\
\text{b) } g(B) \text{ engendre } C_n .
\end{cases}
\]

\[
(68)\qquad F_n : \check{\mathbb{V}}(C_n) \longrightarrow \check{\mathbb{V}}(\underline{O}_S)
\]

\[
(69)\qquad F_n(x) = N_{C_n/\underline{O}_S}(x) \quad \bigl(= N_{\Omega/S}(x, \;)\bigr),
\]

\[
(70)\qquad f_n = \text{\struck{\ill{}}}\, F_n \circ g
\quad \Bigl(\text{dans } \Gamma\, \mathrm{Sym}^n(\check{B} = V)\Bigr)
\]

\[
(71)\qquad C_n \simeq \underline{O}_S[T] / (f)\, \underline{O}_S[T]
\]

\[
f(T) = \text{\struck{\ill{}}}\; T^n + a_1 T^{n-1} + a_2 T^{n-2} + \cdots + a_n
\]

\[
q_n = \text{\struck{\ill{}}}\; T^n + a_{n-1} e_B T^{n-1} + \cdots + a_n e_B^n
\in \Gamma\, \mathrm{Sym}^n B
\]

\[
(74)\qquad N_{C_n/\underline{O}_S}(u 1_C + v t)
= u^n - a_1 u^{n-1} v + \cdots + (-1)^n a_n v^n
= (-v)^n f\Bigl(-\frac{u}{v}\Bigr)
\]

\[
(75)\qquad N_{C_n/\underline{O}_S}(t) = (-1)^n a_n
\]

\[
\begin{pmatrix}
0 & 0 & \cdots & 0 & -a_n \\
1 & 0 & \cdots & 0 & -a_{n-1} \\
0 & 1 & \cdots & 0 & \vdots \\
\vdots & \vdots & & \vdots & \vdots \\
0 & 0 & \cdots & 1 & -a_1
\end{pmatrix}
\]

\[
(-1)^{n-1}(-a_n) = (-1)^n a_n
\]

\[
(76)\qquad f(T) = \prod_{i=1}^{n} (T - t_i)
\qquad \text{donc } a_i = (-1)^i \sigma_i(t_1, \ldots, t_n)
\]

\[
N_{C_n/\underline{O}_S}(u 1_C + v t) = (-v)^n f\Bigl(1 - \frac{u}{v}\Bigr)
= \prod_i (u + v t_i)
= \prod_i \rho_i(u 1_C + v t)
\]

\[
f_n = \varepsilon (-1)^n \bigl( e_B'^{\,n} - a_1 e_B'^{\,n-1} T' + \cdots
+ (-1)^i a_i e_B'^{\,n-i} T'^{\,i} + \cdots + (-1)^n a_n T'^{\,n} \bigr),
\]

\[
(73)\qquad f_n(u e_B + v T) = \varepsilon \bigl( u^n - a_1 u^{n-1} v + \cdots
+ (-1)^i a_i u^{n-i} v^i + \cdots + (-1)^n a_n v^n \bigr)
\]

\[
(77)\qquad \rho_i : C_n \longrightarrow \underline{O}_S
\]

\[
(78)\qquad \rho_i(T) = t_i ,
\]

\[
(79)\qquad \Bigl[\Bigl[\; N_{C(t_1, \ldots, t_n)/\underline{O}_S}(w)
= \prod_{1 \leqslant i \leqslant n} \rho_i(w) \;\Bigr]\Bigr]
\]

\[
(80)\qquad \underline{b}(q) \in \Gamma(2F) \subset \Gamma(\check{B} = V)
\]

\[
(81)\qquad
\begin{cases}
\underline{b}(q) : B \longrightarrow \underline{O}_S \\
\underline{b}(q)(x) = \varphi(e_B, x) \quad (= \mathrm{tr}_{B/\underline{O}_S}\, x)
\end{cases}
\]

\[
(82)\qquad
\begin{cases}
q_0(e_B) = 0 \\
\varphi_0(e_B, \;\cdot\;) = 0
\end{cases}
\qquad \text{(\struck{$\forall x$} où $\varphi_0$ est la forme bil.\ associée à $q_0$)}
\]

\[
(83)\qquad q_0(x) = c_0\, \pi_1(x)^2 .
\]

\[
(84)\qquad \sigma : u \longmapsto \text{\struck{\ill{}}}\; (\mathrm{tr}\, u)\, e_B - u
\]

\[
(85)\qquad u + \sigma u = (\mathrm{tr}\, u)\, e_B
\]

\[
\sigma u = u + a \otimes \pi_1(u) \qquad \text{i.e. } \sigma = \mathrm{id}_B + a \otimes \pi_1
\]

\[
a \in \Gamma(B \otimes L \simeq V = \check{B}) \text{ \struck{\ill{}}}
\qquad \text{et} \qquad
\pi_1 \in \Gamma\bigl(\check{B} \otimes L^{\vee} \simeq \underline{\mathrm{Hom}}(B, L^{\vee}) \simeq B\bigr)
\]

\[
\sigma^2 = \mathrm{id}_V + 2\, \pi_1 \otimes a + \langle a, \pi_1 \rangle\, \pi_1 \otimes a
= \mathrm{id}_V + \pi_1 \otimes a\, \bigl(2 + \langle a, \pi_1 \rangle\bigr)
\]

\[
a(e_B) = -2 ,
\]

\[
(86)\qquad u \cdot \sigma u = N(u)\, e_B = q(u)\, e_B
\]

\[
u(\mathrm{tr}\, u - u) = N(u)
\]

\[
(87)\qquad u^2 - u\, \mathrm{tr}\, u + N(u) = 0 \qquad \text{\uncertain{ie} \ill{}}
\]

\[
(88)\qquad s_0 = \tfrac{1}{2}\, \text{\struck{$\underline{b}$}}\; = \tfrac{1}{2}\, \mathrm{tr}\ldots
\]

\[
(89)\qquad \alpha_{s_0} : \underline{O}_S + L^{\vee} \xrightarrow{\;\sim\;} B
\]

\[
\underline{b}(s_0) \in \Gamma L, \qquad \underline{c}(s_0) \in \Gamma L^{\otimes 2}
\]

\[
(90)\qquad \text{\struck{\ill{}}}\; \underline{b}(s_0) = 0
\]

\[
\text{\struck{$\underline{c}$}}\,(91)\qquad \text{\struck{\ill{}}}\;
\partial_{\xi} = b_0^2 - 4 c_0 = -4 c_0
\]

\[
(92)\qquad \Omega = \tfrac{1}{2}\, \Omega(\partial)
\]

\[
x^2 - b x + c = 0
\]

\[
x^2 = \tfrac{1}{4}\, \partial
\]

\[
(2x)^2 = \partial
\]

\[
\underline{\mathrm{Aff}}(F) \simeq \underline{\mathrm{Aff}}\, \check{\mathbb{V}}(L)
\simeq \underline{\mathrm{Sym}}(L^{\vee}) \simeq \coprod_{n \geqslant 0} L^{\otimes(-n)}
\]

\[
(93)\qquad (\hat{c} + b + 1) \otimes L^{-2} \subset \underline{O}_S + L^{-1} + L^{\otimes(-2)},
\qquad (c + b + 1)^{\wedge} \in \Gamma(L^2 \oplus L \oplus \underline{O}_S)
\]

\[
B \simeq \underline{\mathrm{Sym}}(L^{\vee}) / (1 - 4\partial)\, L^{\otimes -2} ,
\]

\[
(94)\qquad \sigma x = b - x
\]

\[
(95)\qquad 2F \simeq \check{\mathbb{V}}(L)
\]

\[
(96)\qquad \mathrm{tr}(e_B) = 0
\]

\[
(97)\qquad \underline{b} = \mathrm{tr} \in \Gamma(S, L)
\]

\[
(98)\qquad \partial = \underline{b}^2
\]

\[
(99)\qquad \text{\struck{\ill{}}}\; b(s) = \underline{b}
\qquad \text{(\uncertain{indépendant} de $s$ !)}
\]

\[
\partial = b^2 - 4c = b^2 = \underline{b}^2 \qquad (\text{car } 4 = 0)
\]

\[
(100)\qquad \gamma \in H^1(S, L)
\]

\[
F / \underline{b}
\]

\[
(101)\qquad 0 \longrightarrow (\mathbb{Z}/2\mathbb{Z})_S \longrightarrow \check{\mathbb{V}}(L)
\xrightarrow{\;\beta\;} \check{\mathbb{V}}(L^{\otimes 2}) \longrightarrow 0
\]

\[
\varepsilon \longmapsto \varepsilon\, \underline{b}, \qquad x \longmapsto x^2 + \underline{b}\, x
\]

\[
(102)\qquad F / \underline{b} \simeq F \wedge^{(\check{\mathbb{V}}(L), \beta)} \check{\mathbb{V}}(L^{\otimes 2})
\overset{\mathrm{dif}}{=} C
\]

\[
(103)\qquad \partial \in \Gamma(S, L^{\otimes 2})
\]

\[
(104)\qquad x^2 = \partial \qquad \bigl(x \in \Gamma(-, L)\bigr)
\]

\[
i : S_f \longrightarrow S' \qquad (f = 2 \cdot 1_{S'})
\]

\[
\text{\struck{\ill{}}}\quad \underline{L} \subset \tfrac{1}{f}\, \underline{L} \subset i_*(L|S_f)
\]

\[
\tfrac{1}{f}\,\underline{L} \,/\, \underline{L} \;\simeq\; \underline{L} / f\,\underline{L} ,
\]

\[
b_0 = s + \tfrac{1}{f}\, x \qquad \bigl(x \in \Gamma(S, L)\bigr)
\]

\[
S'_0 = V(f) = V(f\,\underline{O}_S) = V(2\,\underline{O}_S),
\]

\[
(105)\qquad b_0 \in \Gamma(S_0, L_0)
\]

\[
f x \,|\, S_0 = b_0 .
\]

\[
(106)\qquad b_0^2 = \partial_0 \quad (= \partial \,|\, S_0)
\]

\[
\Theta : q \longmapsto \text{\struck{\ill{}}} = \partial(q) = -\operatorname{discr}(q)
\]

\[
(107)\qquad \Theta(q + \varpi) = \Theta(q) - 4\varpi \qquad \bigl(\varpi \in \Gamma(L^{\otimes 2})\bigr)
\]

\[
(108)\qquad \Theta_0(\varpi) = -4\varpi \qquad \text{i.e. } \Theta_0 = -4\, \mathrm{id}
\]

\[
(109)\qquad \Theta(q) = \partial
\]

\[
(110)\qquad q = q_0 + \varpi, \qquad \varpi \in \Gamma(S, L^{\otimes 2})
\]

\[
(111)\qquad \Theta(q_0) - 4\varpi = \partial \qquad \text{i.e. } 4\varpi = \Theta(q_0) - \partial
\]

\[
(112)\qquad S_1 = V(4\,\underline{O}_S) \subset S
\]

\[
(113)\qquad a_1(\partial) = a_1 \in \Gamma(S_1, L^{\otimes 2}),
\]

\[
(114)\qquad a_1(\partial) = 0
\]

\[
\exists\, \Omega_U \text{ sur } U \text{ \struck{satisfaisant}} \text{ \add{correspondant} aux }
\text{\struck{\ill{}}} \text{ \add{données}}
\]

\[
(\text{\uncertain{K.1}})\qquad b_0^2 = \partial_0
\]

\[
\Omega \longmapsto (L, \partial, b_0)
\]

\[
\partial_1 = \gamma(b_0),
\]

\[
\gamma : \text{\struck{\ill{}}}\; \underline{O}_{S_0} \longrightarrow \underline{O}_{S_1}
\]

\[
\underline{O}_{S_1} \longrightarrow \underline{O}_{S_1}, \qquad \beta_1 \longmapsto \beta_1^2 .
\]

\[
\Omega(b, c) \subset \underline{L}, \qquad \Omega(b, c) = \{\, x \mid x^2 - bx + c = 0 \,\}
\]

\[
\begin{cases}
b' = b + 2\tau \\
c' = c + \tau b + \tau^2
\end{cases}
\]

\[
b^2 - 4c = b'^{\,2} - 4c'_1 = \partial
\]

\[
4(c'_1 - c') = 0
\]

\[
b_{S_0} = b_0, \qquad b^2 - 4c = \partial .
\]

\[
  \vartheta_1 = \gamma(b_0)
\]

\[
  b_1^2 = \vartheta_1 .
\]

\[
  \text{\struck{$\beta(\alpha(\vartheta_1)) = \gamma(b_0) - \vartheta_1$}}
\]

\[
  \text{\struck{$\beta : L_0 \xrightarrow{\ \sim\ } \operatorname{Ker}(L_1 \to L_0)$}}
\]

\[
  b_1^2 = \vartheta_1 .
\]

\[
  b^2 - 4c = \vartheta .
\]

\[
  \alpha(\vartheta_1) = 0
\]

\[
  \alpha(\vartheta_1) \in \Gamma(S_0, L_0)
\]

\[
  \beta(\alpha(\vartheta_1)) = \gamma(b_0) - \vartheta_1
\]

\[
  \beta : L_0 \xrightarrow{\ \sim\ } \operatorname{Ker}(L_1 \to L_0)
\]

\[
  A_I \xrightarrow{\ \varphi\ } AQ_I
\]

\[
  \varphi(X_{ij}) =
  \begin{cases}
    Y_{\{i,j\}} & \text{si } i \neq j \\
    2Y_{\{i,j\}} = 2X_{\{i\}} & \text{si } i = j
  \end{cases}
\]

\[
  MS_I(k) \longrightarrow M_I(k)
\]

\[
  \Delta_0 \in A_I
\]

\[
  \Delta_0 = \sum_{\sigma \in \mathfrak{S}_I} \varepsilon_\sigma
  \underbrace{\prod_{i \in I} X_{i, \sigma i}}_{X_\sigma}
\]

\[
  X_\sigma = \prod_{i \in I_1(\sigma)} X_{ii}
  \prod_{i \in I_2(\sigma)} X_{\{i, \sigma i\}}
  \prod_{i \in I_*(\sigma)} X_{\{i, \sigma i\}}
\]

\[
  \varepsilon_\sigma = (-1)^{\nu_2 + \nu_*^+}
\]

\[
  \Delta = \sum_{\sigma \in \mathfrak{S}_I} (-1)^{\nu_2 + \nu_*^+} 2^{\nu_1}
  \underbrace{\prod_{i \in I_1(\sigma)} Y_{\{i\}}
  \Bigl(\prod_{H \in \dot I_2(\sigma)} Y_H\Bigr)^2
  \prod_{H \in \dot I_*(\sigma)} Y_{H,\sigma}}_{Y_\sigma}
\]

\[
  Y_{H,\sigma} = \prod_{a \in A(H,\sigma)} X_a ,
\]

\[
  I_1(\sigma) \sqcup I_2(\sigma) \sqcup I_3(\sigma),
\]

\[
  Y_\sigma = \prod_{i \in I_1(\sigma)} Y_i
  \Bigl(\prod_{a \in I_2(\sigma)/\sigma_2} Y_a\Bigr)^2
  \prod_{a \in A(I_3(\sigma))} Y_a
  \;=\; Y_{\gamma(\sigma)}
\]

\[
  \Delta = \sum_{\gamma \in \Gamma_I} (-1)^{\nu_2 + \nu_*^+(\gamma)}
  2^{\nu_1 + \nu_*(\gamma)}
  \underbrace{\prod_{i \in I_1(\gamma)} Y_i
  \Bigl(\prod_{a \in A(I_2(\gamma))} Y_a\Bigr)^2
  \prod_{a \in A(I_3(\gamma))} Y_a}_{M_\gamma}
\]

\[
  \Delta \bmod 2 = \sum_{\gamma \in \Gamma_0}
  \Bigl(\prod_{a \in A(\gamma_0)} Y_a\Bigr)^2
\]

\[
  \operatorname{card} I = m = 2n
\]

\[
  B = B_I = \sum_{\gamma \in \Gamma_0} \Bigl(\prod_{a \in A(I, \gamma_2)} Y_a\Bigr) .
\]

\[
  B^2 - \varepsilon \Delta \equiv 0 \quad (4)
\]

\[
  B^2 - 4C = \varepsilon \Delta .
\]

\[
  B^2 = \sum_{\gamma \in \Gamma_0} \Bigl(\prod_{a \in A(\gamma)} Y_a\Bigr)^2
  + 2 \sum_{\substack{\gamma, \gamma' \in \Gamma_0 \\ \gamma \neq \gamma'}}
  \Bigl(\prod_{a \in A(\gamma)} Y_a\Bigr)\Bigl(\prod_{a' \in A(\gamma')} Y_{a'}\Bigr)
\]

\[
  \Delta \equiv (-1)^n \sum_{\gamma \in \Gamma_0}
  \Bigl(\prod_{a \in A(\gamma)} Y_a\Bigr)^2
  + 2 \sum_{\gamma \in \Gamma_1} (-1)^{\nu_2(\gamma) + 1}
  \qquad \bmod (4)
\]

\[
  B^2 - (-1)^n \Delta \equiv
  \sum_{\substack{\gamma, \gamma' \in \Gamma_0 \\ \gamma \neq \gamma'}}
  \underbrace{\Bigl(\prod_{a \in A(\gamma)} Y_a\Bigr)
  \Bigl(\prod_{a' \in A(\gamma')} Y_{a'}\Bigr)}_{M_{\gamma, \gamma'}}
  \quad \bmod 4
\]

\[
  I_{\sigma, \sigma'} = \{ i \in I \mid \sigma i = \sigma' i \}, \quad
  I = I_{\sigma, \sigma'} \sqcup J_{\sigma, \sigma'} ,
\]

\[
  B^2 - (-1)^n \Delta \equiv
  \sum_{\substack{\gamma \in \Gamma \\ \nu_1(\gamma) = 0 \\ \nu_*(\gamma) = \nu_*^+(\gamma) > 0}}
  2^{\nu_*(\gamma)} M_\gamma
  \quad \bmod 4
\]

\[
  + \, 2 \sum_{\gamma \in \Gamma_1} (-1)^{n + \nu_2(\gamma)} M_\gamma
\]

\[
  \equiv 2 \sum_{\gamma \in \Gamma_1} M_\gamma \quad \not\equiv 0 \quad (4)\,!
\]

\[
  + \, 2 \sum (-1)^{\varepsilon(\gamma)} M_\gamma
\]

\[
  = 2 \sum M_\gamma \bigl(\underbrace{1 + (-1)^{\varepsilon(\gamma)}}_{0 \text{ ou } 2}\bigr)
  \equiv 0 \quad (4)
\]

\[
  \Delta = b^2 - 4c \qquad b, c \in A
\]

\[
  \Delta_0 \in A_0^2 \qquad (\Delta_0 = (\Delta \bmod 2))
\]

\[
  \Delta_0 = b_0^2 \qquad b_0 \in A_0
\]

\[
  b'^2 = b^2 + 4(x^2 + bx) \qquad \text{i.e.} \qquad b'^2 \equiv b^2 \quad (4)
\]

\[
  \Delta_0 v_0^2 = b_0'^2
\]

\[
  \Bigl(\frac{b'_0}{v_0}\Bigr)^2 = \Delta_0 \qquad \text{relation dans } A_{0\Delta_0}
\]

\[
  b'_0 = b_0 v_0 \qquad (b_0 \in A_0,\ b_0^2 = \Delta_0) .
\]

\[
  \gamma(b'_0) \equiv b'^2 \pmod 4 = b^2 v^2
\]

\[
  b^2 v^2 \equiv \Delta v^2 \quad (4)
\]

\[
  b^2 \equiv \Delta \quad (4)
\]

\[
  \begin{cases}
    b'^2 - 4c' = \Delta^{\nu} v^2 \\
    v_0 \text{ est } A_0\text{-régulier}
  \end{cases}
\]

\[
  b^2 - 4c = \Delta .
\]

\[
  A_{2\Delta} = (A_2)_\Delta = (A_\Delta)_2 , \quad \text{localisé de } A \text{ pour } 2\Delta
\]

\[
  B_{2\Delta} = A_{2\Delta}[T] / (T^2 - \Delta)
\]

\[
  B_\Delta \simeq A_\Delta[T] / (T^2 + bT + c) \qquad b, c \in A_\Delta
\]

\[
  \mathfrak{D}(B_\Delta / A_\Delta) = (b^2 - 4c) A_\Delta
\]

\[
  b^2 - 4c \in A_\Delta^* .
\]

\[
  b^2 - 4c = u^2 \Delta , \qquad u \in A_{2\Delta}^* = (A_\Delta)_2^*
\]

\[
  \text{\struck{$b^2 - 4c = 2^{2\mu} \Delta^{2\nu + 1} v^2$ dans $(A_{2\Delta})_2$}}
\]

\[
  b^2 - 4c = \Delta u^2 \qquad u \in A^*
\]

\[
  \text{\struck{$b^2 - 4c = \lambda^2 \Delta \quad \lambda \in A^* \quad (\lambda = (\Delta^\nu v)^2)$}}
\]

\[
U=\frac{1}{2u}\Bigl(T+\frac{b}{2}\Bigr).
\]

\[
b^2-4c=\Delta^{\nu}u^2 .
\]

\[
\text{\struck{$A_\Delta^{*}=\{u\Delta^{d}\mid d\in\mathbf{Z},\ u\in A^{*}\}$}}
\]

\[
\text{\struck{$b^2-4c=\Delta^{2d+1}u^2\quad (d\in\mathbf{Z},\ u\in A^{*})$}}
\]

\[
\begin{cases}
\operatorname{Ker}\bigl(\operatorname{Pic}(A_\Delta)\to\operatorname{Pic}(A_{2\Delta})\bigr)=0, \\
A_\Delta \text{ réduit et } \dots
\end{cases}
\]

\[
b^2-4c=\Delta
\]

\[
A_{2\Delta}[U]/(U^2-\Delta)\qquad [\text{NB. } A[T]/(T^2+bT+c)]
\]

\[
\text{\struck{$\begin{cases} 2\in A \text{ régulier} \\ A_0=A/2A \text{ intègre normal} \\ A \text{ noeth.} \\ \Delta\in A,\quad \Delta\notin 2A \end{cases}$}}
\]

\[
b^2-4c=\Delta
\]

\[
\begin{cases}
X\supset U=X-D \\
X_0=V(2\mathcal{O}_X)\qquad X_0 \text{ intègre normal}
\end{cases}
\]

\[
\ U-U_0=U\cap V\qquad V=X-X_0,\qquad U_0=U\cap X_0 .
\]

\[
L_{U'}^{\,\otimes 2}\xrightarrow{\ \sim\ }\underline{\mathcal{O}}_{U'}
\]

\[
\begin{cases}
E_U \text{ fibré sur } L_U \\
L_U^{\otimes 2}\ \text{\struck{$\simeq\mathcal{O}_S$}}\ \Delta
\end{cases}
\]

\[
U\cup V=W=X\smallsetminus \underbrace{X_0\cap D}_{D_0}
\]

\[
\widetilde{V}_1\to V_1\subset L
\]

\[
\tfrac{1}{2}L\qquad\qquad
\begin{cases} 2x\in L \\ x^2 \end{cases}
\]

\[
\begin{array}{lccc}
2^4\,Y_1Y_2Y_3Y_4 & \text{(quatre points)} & 1 & 1 \\
-2^2\,Y_1Y_2(Y_{3,4})^2 & \text{(deux points, une arête)} & 6 & 6 \\
2^2\,Y_1\,Y_{23}Y_{34}Y_{41} & \text{(un point, un triangle)} & 4 & 8 \\
(Y_{12}Y_{34})^2 & \text{(deux arêtes disjointes)} & 3 & 3 \\
-2\,Y_{12}Y_{23}Y_{34}Y_{41} & \text{(un carré)} & 3 & 6 \\
 & & & 24=4!
\end{array}
\]

\[
\Bigl(\sum Y_{12}Y_{34}\Bigr)^2=\sum
\]

\[
(-1)\sum_{\gamma\in\Gamma_0}M_\gamma
\ +\ (-1)^{\nu_2(\gamma)+1}\,2\!\!\sum_{\substack{\gamma\in\Gamma\\ \nu^{*}(\gamma)=1\\ \nu_1(\gamma)=0}}\!\! M_\gamma
\]

\[
(-1)^n\Delta=\sum_{\gamma\in\Gamma_0}M_\gamma+(-1)^{n+\nu_2(\gamma)}\sum_{\gamma\in\Gamma_1}M_\gamma
\]

\[
(-1)^n\Delta-B^2\equiv 2\sum_{\gamma\in\Gamma_1}(-1)^{n+1+\nu_2(\gamma)}M_\gamma
\ -\underbrace{\sum_{\substack{\gamma,\gamma'\in\Gamma_0\\ \gamma\neq\gamma'}}M_\gamma M_{\gamma'}}_{2\sum_{\gamma\in\Gamma_1}M_\gamma}
\]

\[
=2\sum_{\gamma\in\Gamma_1}\bigl[(-1)^{(\dots)}-1\bigr]M_\gamma
\]

\[
\delta(q)=-\omega^{\otimes 2}\qquad \bigl(\omega\in\Gamma(S,(\det E)^{*})\bigr) \tag{1}
\]

\[
E\xleftarrow{\ \sim\ }\gamma\mathcal{M}=\operatorname{Ker}\bigl(\mathcal{M}\xrightarrow{\ \mathrm{tr}\ }\underline{\mathcal{O}}_S\bigr) \tag{2}
\]

\[
q_{\mathcal{M}}(u)=-\det u\qquad \bigl(u\in\Gamma(-,\gamma\mathcal{M})\bigr) \tag{3}
\]

\[
\varphi_{\mathcal{M}}([u,v],w)\,\omega_{\mathcal{M}}=2\,u\wedge v\wedge w\qquad \bigl(u,v,w\in\Gamma(-,\gamma\mathcal{M})\bigr) \tag{4}
\]

\[
\begin{cases}
q_{\mathcal{M}}(u+v)=\text{\struck{$\det$}}\ q_{\mathcal{M}}(u)+q_{\mathcal{M}}(v)+\varphi_{\mathcal{M}}(u,v) \\
\text{i.e.}\quad \det(u+v)=\det(u)+\det(v)-\varphi_{\mathcal{M}}(u,v)
\end{cases} \tag{5}
\]

\[
\varphi_{\mathcal{M}}(u,v)=\operatorname{Tr}uv\qquad u,v\in\Gamma(-,\gamma\mathcal{M}) \tag{6}
\]

\[
\text{\struck{$\operatorname{Tr}([u,v]w)\,\omega$}}\quad 2\,u\wedge v\wedge w=\operatorname{Tr}([u,v]w)\,\omega_{\mathcal{M}} \tag{7}
\]

\[
\Bigl(=\operatorname{Tr}(uvw-vuw)\,\omega_{\mathcal{M}},\qquad \operatorname{Tr}(vuw)=-\operatorname{Tr}uvw\Bigr)
\]

\[
u\wedge v\wedge w=\operatorname{Tr}(uvw)\,\omega_{\mathcal{M}}\qquad u,v,w\in\Gamma(-,\gamma\mathcal{M}). \tag{8}
\]

\[
2\,x\wedge y\wedge z=\varphi([x,y],z)\,\omega\qquad \bigl(x,y,z\in\Gamma(-,E)\bigr) \tag{8}
\]

\[
q(x+y)=q(x)+q(y)+\varphi(x,y). \tag{9}
\]

\[
\ \Sigma_q\subset\check{\mathbb{V}}(E)\quad\text{d'équation } q(x)=-1 \tag{10}
\]

\[
\Sigma_{\mathcal{M}}\subset\check{\mathbb{V}}(\underbrace{\gamma\mathcal{M}}_{\mathcal{V}})\quad\text{d'équation } q_{\mathcal{M}}(u)=-1\quad\text{i.e.}\quad \det u=1 \tag{11}
\]

\[
\operatorname{Sym}^{*}(\check{E})/(q+1)\operatorname{Sym}^{*}(\check{E})\simeq\operatorname{Sym}^{*}\bigl((\gamma\mathcal{M})^{\vee}\bigr)/(q_{\mathcal{M}}+1) \tag{12}
\]

\[
S[\underline{i}]=\operatorname{Spec}\underbrace{\underline{\mathcal{O}}_S[\underline{i}]}_{=\underline{\mathcal{O}}_S[T]/(T^2+1)}, \tag{13}
\]

\[
\Sigma\hookrightarrow X^{S[\underline{i}]}=\underline{\operatorname{Hom}}_S(S[\underline{i}],X) \tag{14}
\]

\[
\begin{cases}
X\simeq\widetilde{X}/\mathbb{G}_m \\
\widetilde{X}\subset\check{\mathbb{V}}(E)^{*}\qquad \widetilde{X}\ \text{d'équation } q(x)=0
\end{cases} \tag{15}
\]

\[
\xi=x+\underline{i}y\qquad x,y\in\Gamma(S,E),
\]

\[
\begin{cases} q(x)=q(y) \\ \varphi(x,y)=0 \end{cases} \tag{17}
\]

\[
\underbrace{(a+\underline{i}b)}_{\alpha}\underbrace{(x+\underline{i}y)}_{\xi}=(ax-by)+\underline{i}(bx+ay)
\]

\[
\ q(x)=q(y)\in\Gamma(S,\mathbb{G}_m) \tag{18}
\]

\[
q(\alpha\xi)=q(ax-by)=a^2q(x)+b^2q(y)-2ab\underbrace{\varphi(x,y)}_{0}=(a^2+b^2)\,q(x)\in\Gamma\mathbb{G}_m
\]

\[
\mathcal{U}=\widetilde{\mathcal{U}}/\mathbb{G}_m^{S[\underline{i}]}\subset X^{S[\underline{i}]} \tag{19}
\]

\[
\widetilde{\mathcal{U}}\longrightarrow\ \check{\mathbb{V}}(E)
\]

\[
\xi=x+\underline{i}y\overset{\tilde f}{\longmapsto}\ q(x)^{-1}[x,y] \tag{20}
\]

\[
q([x,y])=-4q(x)q(y)\underbrace{\sin^2(\widehat{x,y})}_{\overset{\text{déf}}{=}\ \bigl(1-\frac{1}{4}\varphi(x,y)^2/q(x)q(y)\bigr)} \tag{21}
\]

\[
\text{\struck{$4$}}\ q([x,y])=-4q(x)q(y)+\varphi(x,y)^2 \tag{21 bis}
\]

\[
\text{\struck{$2\varphi([x,y],[x,y])=4(x\wedge y\wedge[x,y])$}}
\]

\[
\ q([x,y])=-4q(x)^2
\]

\[
q\Bigl(\frac{\text{\struck{$4$}}\,[x,y]}{2q(x)}\Bigr)=-1
\]

\[
\tilde f:\widetilde{\mathcal{U}}\longrightarrow\Sigma_{1} \tag{22}
\]

\[
\tilde f(\underbrace{x+\underline{i}y}_{\xi})=[x,y]/2q(x)
\]

\[
\tilde f(\alpha\xi)=\tilde f(\xi).
\]

\[
\alpha\xi=(ax-by)+\underline{i}(bx+ay)=x'+\underline{i}y'
\]

\[
[x',y']=[ax-by,\,bx+ay]=(a^2+b^2)[x,y]
\]

\[
q(x')=q(y')=q(ax-by)=(a^2+b^2)\,q(x)
\]

\[
[x',y']/2q(x')=[x,y]/2q(x).
\]

\[
\mathcal{U}\xrightarrow{\ f\ }\Sigma,\qquad \mathcal{U}\subset X^{S[\underline{i}]} \tag{23}
\]

\[
[x,y]/2q(x)=[x',y']/2q(x'),
\]

\[
\begin{cases} x'=ax-by \\ y'=cx+dy \end{cases}\qquad\text{avec } ad-bc\in\Gamma\mathbb{G}_m
\]

\[
\varphi(x',y')=ac\,\varphi(x,x)-bd\,\varphi(y,y)=0\quad\text{i.e. comme}\quad \varphi(x,x)=2q(x)=\varphi(y,y)=2q(y)
\]

\[
ac-bd=0\qquad\qquad \begin{matrix} a & b\\ d & c\end{matrix}
\]

\[
\begin{cases}
q(x')=(a^2+b^2)\,q(x)\\
q(y')=(c^2+d^2)\,q(y)=\lambda^2(a^2+b^2)\,q(x)
\end{cases}
\]

\[
\text{\struck{$(\lambda^2-1)(a^2+b^2)=0$}}
\]

\[
\begin{cases} a^2+b^2\in\Gamma\mathbb{G}_m \\ \lambda^2=1 \end{cases}
\]

\[
x'\wedge y'/2q(x')=x\wedge y/2q(x)
\]

\[
\begin{cases}
x'\wedge y'=(ad+bc)\,x\wedge y=\lambda(a^2+b^2)\,x\wedge y\\
q(x')=(a^2+b^2)\,q(x)
\end{cases}
\]

\[
x'\wedge y'/q(x')=\lambda\,x\wedge y/q(x)
\]

\[
\xi'=\alpha\xi,\quad \alpha=a+\underline{i}b,\quad a^2+b^2\in\Gamma\mathbb{G}_m,
\]

\[
q(x)=q(y)=1,\quad \varphi(x,y)=0
\]

\[
t'=\tilde f(x+\underline{i}y)=x\wedge y/q(x)=x\wedge y
\]

\[
\text{Donc}\quad
\begin{cases}
q(x)=q(\lambda y)=1,\quad \varphi(x,\lambda y)=0\\
\text{\struck{$x+\underline{i}\lambda y$}}\ \tilde f(x+\underline{i}\lambda y)=x\wedge\lambda y=\lambda\,x\wedge y=\lambda t'=t
\end{cases}
\qquad \text{OK.}
\]

\[
\Sigma\overset{g}{\hookrightarrow}X^{S[\underline{i}]} \tag{24}
\]

\[
\text{\struck{$\delta(q')=-\delta(q)=\omega^{\otimes 2}$}}
\]

\[
\text{\struck{$\varphi'(x,y)=-\varphi(x,y)$}}
\]

\[
\text{\struck{$\varphi'([x,y]',z)\,\omega=2\,x\wedge y\wedge z$}}
\]

\[
\text{\struck{$\varphi([x,y],z)\,\omega=-\varphi'([x,y],z)\,\omega$}}
\]

\[
\text{\struck{$[x,y]'=-[x,y]$}}
\]

\[
q'([x,y]')=4q'(x)q'(y)\underbrace{\sin^2(\widehat{x,y})'}_{1-\varphi'(x,y)^2/4q'(x)q'(y)} \tag{25}
\]

\[
X\hookrightarrow X^{S[\underline{i}]}
\]

\[
\widetilde{X}\longrightarrow\widetilde{X}^{S[\underline{i}]},\qquad x\longmapsto x+\underline{i}\,0
\]

\[
\text{\struck{$x=ax_0,\quad y=bx_0$}}
\]

\[
\boxed{\ \Sigma\xrightarrow{\ \sim\ }\widetilde{X}^{S[\underline{i}]}-\widetilde{X}=\mathcal{U}\ }
\]

\[
x+\underline{i}y=(a+\underline{i}b)\,x_0,
\]

\[
\text{\struck{$x+\underline{i}y=\underbrace{(x+y)}_{z}+\varepsilon y\longmapsto[x,y]/q(x)$}}
\]

\[
\text{\struck{$=[x+y,y]/q(y)$}}\qquad \text{\struck{$(x+y\neq 0\ \text{i.e.}\ x\neq y)$}}
\]

\[
\text{\struck{$z+\varepsilon y\longmapsto[z,y]/q(y)$}}
\]

\[
\text{\struck{$z\neq 0,\quad q(y+z)=q(y)\neq 0,\quad q(z)=\varphi(y,z)$}}
\]

\[
\text{\struck{$\det u=1,\ \operatorname{Tr}u=0\quad(\Rightarrow u^2+1=0)$}}
\]

\[
\text{\struck{$x+\underline{i}y:\ q(x)=q(y)\neq 0,\ \varphi(x,y)=0,\quad y\neq\lambda$}}
\]

\[
\text{\struck{$?\Leftrightarrow\ u=[v,w]/\det w$}}
\]

\[
\text{\struck{si $\det v=\det w\neq 0$, $\operatorname{Tr}v=\operatorname{Tr}w=0$, $\operatorname{Tr}vw=0$, $v\neq w$}}
\]

\[
\text{\struck{$\det[v,w]$}}
\]

\[
\delta(q'')=-2^6\,\delta(q)=2^6\,\omega^{\otimes 2}=(8\omega)^{\otimes 2}=\omega''^{\otimes 2}
\]

\[
\delta(q'')=\omega''\cdot\omega''
\]

\[
q''(\ \dots \tag{26}
\]

\[
q_\lambda=\lambda q\qquad \varphi_\lambda=\lambda\varphi\qquad [\ ,\ ]_{\lambda\mu}=(\lambda\mu)^{-1}[\ ,\ ],\qquad \omega_\mu=\mu\omega
\]

\[
\text{\struck{$q_\lambda([x,y]_\lambda)=\lambda$}}
\]

\[
\text{\struck{$2\,x\wedge y\wedge z=\varphi_\lambda([x,y]_\lambda,2)\,\omega$}}
\]

\[
\begin{cases}
\delta(q_\lambda)=\lambda^3\delta(q)=-\lambda^3\,\omega^{\otimes 2}=-\lambda^3\mu^{-2}\,\omega_\mu^{\otimes 2}\\
2\,x\wedge y\wedge z=\varphi_\lambda([x,y]_{\lambda,\mu},z)\,\omega_\mu
\end{cases}
\]

\[
\begin{cases}
q_\lambda([x,y]_{\lambda,\mu})=\lambda(\lambda\mu)^{-2}q([x,y])\\
\qquad =-4\lambda^{-1}\mu^{-2}q(x)q(y)\sin^2(\widehat{x,y})\\
\qquad =-4\lambda^{-3}\mu^{-2}q_\lambda(x)q_\lambda(y)\sin^2(\widehat{x,y})
\end{cases}
\]

\[
\lambda=-2^{\alpha},\ \mu=2^{\beta}\qquad \lambda^{-3}\mu^{-2}=-2^{-(3\alpha+2\beta)}=-2^{2}
\]

\[
3\alpha+2\beta=-2\qquad \text{p.ex.\ on prend } \alpha=0,\ \beta=-1
\]

\[
\lambda=-1,\quad \mu=\tfrac12\qquad q_\lambda=q'=-q,\quad \omega_\mu=\omega'=\tfrac12\omega
\]

\[
\begin{cases}
\delta'(q')=+4\,\omega'^{\otimes 2}=(2\omega')^{\otimes 2}\\
x\wedge y\wedge z=\frac12\,\varphi'([x,y]',z)\,\omega'\\
q'([x,y]')=q'(x)q'(y)\sin^2(\widehat{x,y})\\
\qquad =q'(x)q'(x)\bigl(1-\tfrac14\varphi(x,y)^2\ \dots\bigr)
\end{cases}
\]

\[
[x,y]_{\lambda\mu}=[x,y]'=-2[x,y]
\]

\[
  \text{\struck{$(a + \mathbf{i}b)(x + \mathbf{i}y) = \underbrace{ax - by}_{x'} + \mathbf{i}(\underbrace{bx + ay}_{y'})$,}}
\]

\[
  \text{\struck{$x + \mathbf{i}y \in \Gamma\tilde{U} \Longrightarrow [x,y]/q(x) \in \Sigma_{-4}$}}
\]

\[
  \text{\struck{$\tilde{g} : \tilde{U} \longrightarrow \Sigma_{-4}$}}
\]

\[
  g : \text{\struck{$\tilde{X}$}}\ \longrightarrow \Sigma_{-4}
\]

\[
  [x,y] = \varphi(x,y)\, H \quad \text{où} \quad
  \begin{cases}
    H \text{ base du centre de } \tilde{\mathfrak{g}} \\
    q(H) = 1
  \end{cases}
\]

\[
  q([x,y]) = \varphi(x,y)^{2}.
\]

\[
  0 \longrightarrow V \longrightarrow \mathcal{E} \longrightarrow \mathcal{O}_{S} \longrightarrow 1
\]

\[
  \mathcal{E} = V \oplus V^{\perp},
\]

\[
  \Sigma = \check{\mathbb{V}}(q+1) = \{x \in \Gamma(E, V) \mid q(x) = -1\}
\]

\[
  \delta'(q^{\ast}) \in (\det \check{V})^{\otimes 2}
  \qquad \text{base de } (\det \check{V})^{\otimes 2},
\]

\[
  \check{\mathbb{V}}(\det \check{V}) = \mathbb{V}(\det(\check{V})) = \mathbb{V}(\det(V)^{-1})
\]

\[
  \omega^{\otimes(-2)} = \delta'(q).
\]

\[
  \hat{E}^{\sigma} = \sigma(S) \sqcup H, \quad \text{(}H = \mathbb{P}(V)\ \text{hyperplan à l'infini de } \hat{E}\text{)}
\]

\[
  \hat{\Sigma}^{\sigma} = H \cap \hat{\Sigma} = X.
\]

\[
  \mathbf{Hom}_{S}(\Omega, X) \overset{\text{déf}}{=} X^{\Omega} ;
\]

\[
  (X^{\Omega})^{\sigma} \simeq X.
\]

\[
  X^{\Omega} \xrightarrow{\ \sim\ } \hat{\Sigma}
\]

\[
  \Sigma \simeq X^{\Omega} \setminus X,
\]

\[
  X^{\Omega} \xrightarrow{\ \sim\ } \hat{\Sigma}
\]

\[
  (X^{\Omega})_{\Omega} \xrightarrow{\ \sim\ } (\hat{\Sigma})_{\Omega}
\]

\[
  (X^{\Omega})_{\Omega} \simeq (X_{\Omega})^{\Omega_{\Omega}}, \quad
  \Omega_{\Omega} = \Omega \times_{S} \Omega = \Omega \sqcup \Omega,
  \quad \simeq X_{\Omega} \times X_{\Omega},
\]

\[
  X_{\Omega} \times X_{\Omega} \xrightarrow{\ \sim\ } \hat{\Sigma}_{\Omega} \quad (= \widehat{\Sigma_{\Omega}})
\]

\[
  \delta'(q) = \omega^{\otimes 2},
\]

\[
  (\ast) \qquad f_{\omega} : X \times X \longrightarrow \hat{\Sigma}
\]

\[
  f_{-\omega}(x,y) = f_{\omega}(y,x) = \sigma f_{\omega}(x,y),
\]

\[
  \delta'(-q) = -\delta'(q) = -\omega^{\otimes 2},
\]

\[
  E \overset{\varphi}{\simeq} \gamma M
\]

\[
  \operatorname{Tr} u = \det u = 0.
\]

\[
  (X \times X) \setminus X \simeq \Sigma
\]

\[
  \simeq \text{ens.\ des réflexions de } M \quad (\operatorname{tr} = 0,\ \det = -1).
\]

\[
  M \simeq \gamma M \oplus \mathcal{O}_{S},
\]

\[
  \hat{\Sigma}_{M} \subset \check{\mathbb{P}}(M)
\]

\[
  \det(u) + t^{2} = 0 \qquad \text{i.e.} \qquad \det(u + t\,\mathrm{id}) = 0 ;
\]

\[
  F_{M} : X_{M} \times X_{M} \simeq \hat{\Sigma}_{M}, \qquad
  \hat{\Sigma} \cap \mathbb{P}(M) \simeq X
\]

\[
  v \longmapsto v' = \operatorname{tr}(v).1 - v
\]

\[
  v = u + t1
\]

\[
  v' = 2t.1 - (u + t.1) = -u + t1,
\]

\[
  -v' = u - t.1
\]

\[
  \underset{v}{\underbrace{u + t.1}} \longmapsto \underset{-v'}{\underbrace{u - t.1}} .
\]

\[
  f_{-\omega}(x,y) = \sigma f_{\omega}(x,y),
\]

\[
  -\mathrm{id}_{E} : (E, q, \omega) \longrightarrow (E, q, -\omega).
\]

\[
  \delta'(q) = -\omega_{0}^{\otimes 2} \qquad \text{i.e.} \qquad \delta'(-q) = \omega_{0}^{\otimes(-2)}
\]

\[
  X^{\Omega} \setminus X \overset{f_{q,\omega}}{\simeq} \Sigma \qquad (\text{où } \Omega \simeq S[\mathbf{i}])
\]

\[
  \text{\struck{$i^{2} = -1$}}
\]

\[
  \text{\struck{$\omega = i\,\omega_{M}^{\otimes 2}$}}
\]

\[
  \text{\struck{$\delta'(q_{M}) = \omega^{\otimes 2} = -\omega_{M}^{\otimes 2}$}}
\]

\[
  \text{\struck{$(\gamma M, -\det_{\gamma M})$}} \quad \text{\struck{$\delta'(-q_{M}) = -\omega^{\otimes(-2)}$}}
\]

\[
  \text{\struck{$(\gamma M, -q_{M} = \det_{\gamma M}, \omega = i\omega_{M}) \longrightarrow (\gamma M', q_{M'}, \omega_{M'})$}}
\]

\[
  \text{\struck{$M' = M$,}} \qquad
  \text{\struck{$\gamma M \xrightarrow{\ \theta\ } \gamma M' = \gamma M$, \quad $x \longmapsto ix$}}
\]

\[
  q_{M'}(\theta u) = q_{M}(iu) = i^{2} q_{M}(u) = -q_{M}(u),
\]

\[
  \det \theta(\omega) = i^{3}\omega = i^{4}\omega_{M} = \omega_{M} = \omega_{M'}.
\]

\[
  f_{q,\omega} = \theta_{q,\omega}\, g_{q}
\]

\[
  (E, -q, \omega = i\omega_{0}) \simeq (\gamma M', q_{M'}, \omega_{M'})
\]

\[
  (E, q, \omega_{0}) = (\gamma M, q_{M}, \omega_{M}),
\]

\[
  (E, -q, \omega) = (\gamma M, -q_{M}, i\omega_{M})
\]

\[
  \alpha = iu - v, \qquad \beta = iu + v,
\]

\[
  2iu = \alpha + \beta, \qquad 2v = -\alpha + \beta,
\]

\[
  \begin{cases}
    4i[u,v] = 2[\alpha,\beta] \quad \text{i.e.} \quad 2i[u,v] = [\alpha,\beta] \\
    -4q(u) = q(\alpha) + q(\beta) + \varphi(\alpha,\beta) = \operatorname{tr} \alpha\beta \\
    4q(v) = q(\alpha) + q(\beta) - \varphi(\alpha,\beta) = -\operatorname{tr} \alpha\beta
  \end{cases}
\]

\[
  [u,v]/2q(u) = \frac{1}{2i} \cdot \frac{-2}{\operatorname{tr} \alpha\beta}\,[\alpha,\beta]
  = \frac{i}{\operatorname{tr} \alpha\beta}\,[\alpha,\beta] \quad \ldots
\]

\[
  \theta\bigl([u,v]/2q(u)\bigr) = i\,\frac{[u,v]}{2q(u)} = -[\alpha,\beta]/\operatorname{tr} \alpha\beta
\]

\[
  X_{M} \times X_{M} \longrightarrow \hat{\Sigma}_{M}
\]

\[
  \operatorname{tr}(\alpha\beta) \neq 0 \qquad \text{fibre par fibre}),
\]

\[
  -[\alpha,\beta]/\operatorname{tr}(\alpha\beta)
\]

\[
  -[\alpha,\beta]/\operatorname{tr}(\alpha\beta) + 1 \quad \text{ou encore}
\]

\[
  \operatorname{tr}(\alpha\beta).1 - [\alpha,\beta].
\]

\[
  X_{M} \times X_{M} \longrightarrow \hat{\Sigma}_{M}
\]

\[
  D \simeq \mathbb{V}_{\mathrm{aff}}(q_{M})
\]

\[
  \text{\struck{$(\alpha,\beta) \longmapsto$}}
\]

\[
  F_{M} : C_{M} \times C_{M} \longrightarrow D_{M}
\]

\[
  F_{M}(\alpha,\beta) = \operatorname{tr}(\alpha\beta).1 - [\alpha,\beta]
\]

\[
  \mathbb{G}_{m} \times \mathbb{G}_{m} \longrightarrow \mathbb{G}_{m}, \qquad (\lambda,\mu) \longmapsto \lambda\mu
\]

\[
  \Longrightarrow\ q_{M}\bigl([\alpha,\beta]/\varphi(\alpha,\beta)\bigr) = 1
  \qquad \text{i.e.} \qquad \det([\alpha,\beta]) + \operatorname{tr}(\alpha\beta)^{2} = 0
\]

\[
  q_{M}[\alpha,\beta] = \varphi(\alpha,\beta)^{2}
\]

\[
  q_{M}([\alpha,\beta]) = -4\,q(\alpha)\,q(\beta) + \varphi(\alpha,\beta)^{2}
\]

\[
  (1) \qquad \det([\alpha,\beta]) + (\operatorname{tr}(\alpha\beta))^{2} - 4(\det\alpha)(\det\beta) = 0
\]

\[
  u = \alpha + s.1, \qquad v = \beta + t.1 \qquad \text{donc} \qquad
  \operatorname{tr} u = 2s, \quad \operatorname{tr} v = 2t,
\]

\[
  (4) \qquad [u,v] = [\alpha,\beta]
\]

\[
  \text{\struck{$\operatorname{tr}(\alpha\beta) = \operatorname{tr}((u-s)(v-t)) = \operatorname{tr}(uv) + 2st$}}
\]

\[
  \text{\struck{$\operatorname{tr}\alpha\beta = \operatorname{tr} uv - s\operatorname{tr} v - t\operatorname{tr} u + 2st$}}
\]

\[
  \text{\struck{$= \operatorname{tr} uv - \operatorname{tr} u \operatorname{tr} v$}}
\]

\[
  uv = \alpha\beta + s\beta + t\alpha + st \qquad \text{donc}
\]

\[
  \operatorname{tr} uv = \operatorname{tr}\alpha\beta + 2st = \operatorname{tr}\alpha\beta + \tfrac{1}{2}\operatorname{tr} u \operatorname{tr} v
\]

\[
  \operatorname{tr}\alpha\beta = \operatorname{tr} uv - \tfrac{1}{2}\operatorname{tr} u \operatorname{tr} v
\]

\[
  (2) \qquad (\operatorname{tr}\alpha\beta)^{2} = (\operatorname{tr} uv)^{2} - (\operatorname{tr} uv)(\operatorname{tr} u)(\operatorname{tr} v) + \tfrac{1}{4}(\operatorname{tr} u)^{2}(\operatorname{tr} v)^{2}
\]

\[
  (3) \qquad
  \begin{cases}
    \det\alpha = \det(u-s) = \det(s-u) = s^{2} + \det u = \tfrac{1}{4}(\operatorname{tr} u)^{2} + \det u \\
    \det\beta = t^{2} + \det v = \tfrac{1}{4}(\operatorname{tr} v)^{2} + \det v
  \end{cases}
\]

\[
  \det([u,v]) + (\operatorname{tr} uv)^{2} - (\operatorname{tr} uv)(\operatorname{tr} u)(\operatorname{tr} v) + \tfrac{1}{4}(\operatorname{tr} u)^{2}(\operatorname{tr} v)^{2}
\]

\[
  - \tfrac{1}{4}\bigl[(\operatorname{tr} u)^{2} + 4\det u\bigr]\bigl[(\operatorname{tr} v)^{2} + 4\det v\bigr]
\]

\[
  \det([u,v]) - (\det u)(\operatorname{tr} v)^{2} - (\det v)(\operatorname{tr} u)^{2} - 4(\det u)(\det v)
\]

\[
  + (\operatorname{tr} uv)^{2} - (\operatorname{tr} uv)(\operatorname{tr} u)(\operatorname{tr} v) = 0
\]

\[
  (1) \qquad 0 \longrightarrow V \longrightarrow T \xrightarrow{\ \pi\ } \mathcal{O}_{S} \longrightarrow 0
\]

\[
  \check{\mathbb{P}}(\mathrm{Sym}^{2}(\check{T})) = \mathbb{P}(\struck{\ill{}}\,\Gamma^{2}(T)),
\]

\[
  (2) \qquad \delta(q) \in \Gamma(S, (\det T)^{\otimes -2})
\]

\[
  (3) \qquad \mathbb{D} = V^{\perp}
\]

\[
  (4) \qquad \mathbb{D} \simeq (E/V)^{\vee} \simeq \mathcal{O}_{S}^{\vee} \simeq \mathcal{O}_{S}
\]

\[
  (4') \qquad
  \begin{cases}
    a \in \Gamma(S, \mathbb{D}) \subset \Gamma(T) \\
    E = \{x \in \Gamma(-, T) \mid \varphi(x, a) = 1\}
  \end{cases}
\]

\[
  \mathbb{D} \otimes (E/V) \simeq L \quad \text{i.e.}
\]

\[
  (5) \qquad \mathbb{D} \xrightarrow{\ \sim\ } L
\]

\[
  q : \mathbb{D}^{\otimes 2} \longrightarrow L,
\]

\[
  (6) \qquad q : \mathbb{D}^{\otimes 2} \xrightarrow{\ \sim\ } L
\]

\[
  L^{\otimes 2} \simeq L
\]

\[
  (7) \qquad \mathbb{D} \simeq L \simeq \mathcal{O}_{S}
\]

\[
  (8) \qquad \text{a)}\ q(a) = 1
\]

\[
  (9) \qquad \text{b)}\ \text{\struck{$E = \{x \in \Gamma(-,T)$}}\ \varphi(a,x) \text{\struck{$= 1\}$}}
  \quad (x \in \Gamma(-,T))
\]

\[
  (10) \qquad \pi(a) = 2
\]

\[
  \text{\struck{$(11) \qquad \pi(x) = \varphi(a,x)$,}}
\]

\[
  T \simeq V + \mathbb{D} \simeq V + \mathcal{O}_{S} \quad \text{canoniquement, la décomposition}
\]

\[
  (12) \qquad q(x,t) = q_{0}(x) + t^{2}
\]

\[
  q_{0}(x) + 1 = 0 \quad \text{i.e.} \quad q_{0}(x) = -1
\]

\[
  \text{(13)} \qquad \delta'(q) \in \Gamma\bigl(S, (\det T)^{\otimes -2}\bigr)
  \qquad (\text{discr. \struck{divisé}})
\]

\[
  D \xrightarrow{\ \pi \mid D\ } \underline{O}_S
\]

\[
  \text{(14)} \qquad T \simeq V \oplus D .
\]

\[
  \text{(15)} \qquad \pi(a) = 1 \qquad (\ill{} \text{ pour } 2\,!)
\]

\[
  \text{(16)} \qquad q(a) = 1
\]

\[
  \text{(17)} \qquad q(x,t) = q_0(x) + t^2
\]

\[
  \text{(18)} \qquad q_0(x) + 1 = 0 \quad \text{i.e.} \quad q_0(x) = -1
  \qquad \bigl(x \in \Gamma(-,E)\bigr)
\]

\[
  \sigma = \mathrm{id} + v
\]

\[
  v(x) = \pi(x)\cdot c \qquad \text{i.e.} \qquad v = \pi \otimes c
\]

\[
\begin{aligned}
  q \circ \sigma(x) &= q(x + vx) = q\bigl(x + \pi(x)\cdot c\bigr) \\
  &= q(x) + \pi(x)^2 q(c) + \pi(x)\, \varphi(x,c)
\end{aligned}
\]

\[
  q\sigma(x) = \lambda q(x) \qquad \forall x \in \Gamma(-,T)
\]

\[
  (\lambda - 1) q = \pi\bigl(q(c)\pi + \psi(c)\bigr)
  \qquad \bigl(\psi(c) \in \Gamma(S, \check{T}),\
  \psi(c)(x) = \varphi(c,x)\bigr)
\]

\[
  q \circ \sigma = q
\]

\[
  \text{(19)} \qquad q(c)\pi + \psi(c) = 0
\]

\[
  \text{(20)} \qquad \sigma(x) = x - \pi(x)\, a \qquad (\operatorname{rg} E \text{ pair})
\]

\[
  \text{(21)} \qquad \sigma^2 = \mathrm{id}_T
\]

\[
\left\{
\begin{aligned}
  T &= M = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right\} \\
  q &= \det = ad - bc
\end{aligned}
\right.
\qquad (b + b',\ c + c')
\]

\[
  a = \underline{1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}
\]

\[
  \varphi(a,b,c,d;\, a',b',c',d') = bc' + cb' - (ad' + da')
\]

\[
\begin{aligned}
  \pi(a,b,c,d) &= \varphi(a,b,c,d;\, 1,0,0,1) = -(a+d) \\
  &= -\operatorname{Tr}\begin{pmatrix} a & b \\ c & d \end{pmatrix}
\end{aligned}
\]

\[
  V = \gamma M = \left\{ \begin{pmatrix} a & b \\ c & -a \end{pmatrix} \right\}
\]

\[
  \text{(22)} \qquad \sigma(x) = x - 2\pi(x)\, a
\]

\[
  G = \underline{\mathrm{Aut}}(P, H, Q) = \underline{\mathrm{Aut}}_{\mathrm{aff}}(E, Q \cap E)
\]

\[
  G \simeq \underline{\mathrm{Aut}}(V, q_0) \qquad V \text{ de rang pair}
\]

\[
  1 \to G^{+} \to G \to \mathbb{Z}/2\mathbb{Z} \to 1 .
\]

\[
  G \simeq G^{0} \times (\mathbb{Z}/2\mathbb{Z})_S
\]

\[
  G^{0} \xrightarrow{\ \sim\ } \underline{\mathrm{Aut}}^{+}(V, q_0),
  \qquad (\mathbb{Z}/2\mathbb{Z})_S \text{ engendré par } \sigma .
\]

\[
  \text{(1.1)} \qquad \delta'(q) \in \Gamma(\det \check{E})^{\otimes 2}
  = \Gamma(\det E)^{\otimes(-2)}
\]

\[
  \text{(1.2)} \qquad \delta'(q) \in \Gamma\bigl((\det E)^{\otimes(-2)} \otimes L^{\otimes n}\bigr)
  \simeq \underline{\mathrm{Hom}}\bigl((\det E)^{\otimes 2}, L^{\otimes n}\bigr)
\]

\[
  \text{(1.3)} \qquad L^{\otimes n} \simeq (\det E)^{\otimes 2} .
\]

\[
  \text{On pose} \quad
  \left\{
  \begin{aligned}
    &q_n(x_1, y_1, \ldots, x_\nu, y_\nu, z) = \textstyle\sum x_i y_i + z^2 \\
    &\delta'(q_n) = (-1)^\nu
  \end{aligned}
  \right.
\]

\[
  \delta'(-q_n) = -\delta'(q_n) = (-1)^{\nu+1}
\]

\[
\begin{aligned}
  O(q) &= \underline{\mathrm{Aut}}(q) \\
  SO(q) &= \underline{\mathrm{Aut}}(q) \cap Sl(E) = \operatorname{Ker}\bigl(O(q) \xrightarrow{\det} \mathbb{G}_m\bigr)
\end{aligned}
\]

\[
  \text{(2.1)} \qquad \mathbb{G}_m \cap O(q) = \mu_2 \subset O(q)
  \qquad \mu_2 = \underline{\mathrm{Centre}}\, O(q)
\]

\[
  \text{(2.2)} \qquad \mu_2 \cap SO(q) = \{1\}
\]

\[
  \text{(2.3)} \qquad \boxed{\omega^{\otimes -2} = \varepsilon\, \delta'(q)}
\]

\[
  \text{(2.4)} \qquad \underline{\mathrm{Aut}}(E, q, \omega) = SO(q)
\]

\[
\left\{
\begin{aligned}
  &\text{Si } \varepsilon = (-1)^\nu, \text{ alors } q_n \text{ est un prototype} \\
  &\text{Si } \varepsilon = (-1)^{\nu+1}, \text{ — } -q_n \text{ —}
\end{aligned}
\right.
\]

\[
  L = \Delta^{\otimes 2}, \quad E = E_0 \otimes \Delta,
  \quad (q_0 \otimes \Delta^{\otimes 2}) : E_0 \otimes \Delta \to \Delta^{\otimes 2} = L,
\]

\[
  \bigl[\text{donc } \det E \simeq \det E_0 \otimes \Delta^{n}
  \simeq \Delta^{2\nu} \otimes \Delta \simeq L^{\nu} \otimes \Delta\bigr]
\]

\[
  \Delta = L^{\otimes -\nu} \otimes \det E, \quad E_0 = E \otimes \Delta^{-1},
  \quad q_0 = q \otimes \Delta^{-1} : E_0 = E \otimes \Delta^{-1} \to L\Delta^{-2}
\]

\[
  L\Delta^{-2} \simeq L \otimes L^{2\nu} \otimes (\det E)^{-2}
  \simeq L^{n} \otimes (\det E)^{-2} \simeq \underline{O}_S \text{ par (1.3)}
\]

\[
\begin{aligned}
  \det E_0 &\simeq \det E \otimes \Delta^{-n} \\
  &\simeq \det E\, L^{n\nu} (\det E)^{-n} = L^{n\nu} (\det E)^{-2\nu} \\
  &= \bigl(L (\det E)^{-2}\bigr)^{\nu} \simeq \underline{O}_S^{\otimes \nu} \simeq \underline{O}_S
  \qquad \text{etc.\ …}
\end{aligned}
\]

\[
  (E_0, \ldots) \longmapsto \bigl(E_0 \otimes \Delta,\ q_0 \otimes L :
  E_0 \otimes \Delta \to \Delta^{\otimes 2} \xrightarrow[\sim]{\varphi^{-1}} \underline{O}_S\bigr)
\]

\[
\begin{aligned}
  (E, q) \longmapsto \bigl(&\Delta = \det E,\ \varphi = \delta'(q)^{-1},\
  E_0 = E \otimes \Delta^{-1}, \\
  &q_0 = \varepsilon q \otimes \Delta^{-1} : E_0 \to \Delta^{-2}
  \xrightarrow{\ {}^t\varphi\ } \underline{O}_S, \\
  &\omega_0 \in \Gamma(\det E_0) \text{ déduit de }
  \det E_0 \simeq (\det E) \otimes \Delta^{-n} \simeq \underline{O}_S\bigr)
\end{aligned}
\]

\[
  q : E \longrightarrow L
\]

\[
  \text{(4.1)} \qquad q_n = q_{2\nu} = \sum_{1}^{\nu} (x_i y_i)
  \qquad \delta'(q_n) = (-1)^\nu
\]

\[
  \text{(4.2)} \qquad \mu_2 \subset SO(q) \cap Sl(E)
\]

\[
  O(q)/SO(q) \simeq (\mathbb{Z}/2)_S
\]

\[
  \mu_2 = \underline{\mathrm{Centre}}\bigl(O(q)\bigr) \subset SO(q) .
\]

\[
\left\{
\begin{aligned}
  &\underline{\mathrm{Aut}}(X)^{0} \simeq SO(q)/\mu_2 \\
  &\underline{\mathrm{Aut}}(X)/\underline{\mathrm{Aut}}(X)^{0} \simeq (\mathbb{Z}/2)_S
\end{aligned}
\right.
\]

\[
  O(q) \longrightarrow (\mathbb{Z}/2)_S
\]

\[
  \Delta_H(E,q) = \text{ens.\ des drapeaux de type } H \text{ de } E
\]

\[
  E_1 \subset E_2 \subset \cdots \subset E_\nu
  \qquad \bigl(\nu = \operatorname{card} H,\ H = \{\operatorname{rg} E_1, \ldots, \operatorname{rg} E_\nu\}\bigr)
\]

\[
  \underline{\Delta}_H(E,q) : \mathrm{Sch}_{/S} \longrightarrow \mathrm{Ens}
  \qquad S' \longmapsto \Delta_H(E_{S'}, q_{S'})
\]

\[
  \text{\struck{$\Gamma\,\underline{\mathrm{Hom}}(F_0, E_0/F_0 \otimes J)
  \simeq \Gamma\bigl(\underline{\mathrm{Hom}}(F_0, E_0/F_0) \otimes J\bigr)$}}
\]

\[
  \text{\struck{$q(e) = 0,$}} \quad \text{\struck{$\varphi(e, G) = 0$}}
\]

\[
  \text{\struck{$x \mapsto \varphi(e, x) \quad G \to \underline{O}_S$}}
\]

\[
  \text{\struck{$\psi_0 : G_0 \to J$}} \qquad
  \text{\struck{$\psi_0 \in \underline{\mathrm{Hom}}(G_0, J)$}}
\]

\[
  \text{\struck{$e' = e + \xi$}} \qquad \text{\struck{$\xi \in \Gamma(JE)$}}
\]

\[
  \text{\struck{$q(\xi) + \varphi(e, \xi) = 0$}}
\]

\[
\begin{aligned}
  \underline{\mathrm{Hom}}(F_0, E_0/F_0) \otimes J &\longrightarrow \underline{\mathrm{Quad}}(F_0) \\
  u_0 &\longmapsto \bigl(x_0 \mapsto \varphi_0(x_0, u_0 x_0)\bigr)
\end{aligned}
\]

\[
  \underline{\mathrm{Hom}}(F_0, E_0/F_0) \xrightarrow{\ \text{épi}\ }
  \underline{\mathrm{Hom}}(F_0, E_0/F_0^{\perp}) \xrightarrow[\text{épi}]{\ \sim\ }
  \underline{\mathrm{Quad}}(F_0)
\]

\[
  \underline{\Delta}_{\{d,d'\}}(E,q) \longrightarrow \underline{\Delta}_{d}(E,q) .
\]

\[
  \text{\struck{$F^{\perp} \simeq F \oplus G'$}} \qquad E' = F^{\perp}/F
\]

\[
  \underbrace{\det F \otimes \det \check{F}}_{\simeq\, \underline{O}_S} \longrightarrow \det E
\]

\[
  E \simeq F \oplus G \qquad \varphi \mid F \times G \text{ une dualité}
\]

\[
  \varphi(e_i, e_j^{*}) = \delta_{ij}
\]

\[
  \omega_F = e_1 \wedge \cdots \wedge e_n \wedge e_1^{*} \wedge \cdots \wedge e_n^{*}
\]

\[
  q(x \oplus y) = \varphi(x,y) \qquad
  \bigl(= \langle x, y\rangle \text{ si on identifie } G \text{ à } \check{F}\bigr)
\]

\[
  \underline{\mathrm{Hom}}(E/F, F) \longrightarrow \underline{\mathrm{Quad}}(G)
  \text{ est surjectif …}]
\]

\[
  \delta(q) = (-1)^n (\omega_F)^2
\]

\[
  \boxed{(\omega_F)^2 = (-1)^n \delta(q)^{-1}}
\]

\[
  \underline{\Delta}_{\{n\}}(E,q) \longrightarrow M\bigl((-1)^n \delta(q)\bigr)
  = \underline{\mathrm{Arf}}(E,q)
\]

\[
\begin{aligned}
  &\text{\struck{$F' = \underline{O}_S e_1^{*} + \sum_{2 \leqslant i \leqslant n} \underline{O}_S e_i$}} \\
  &\text{\struck{$G' = \underline{O}_S e_1 + \sum_{2 \leqslant i \leqslant n} \underline{O}_S e_i^{*}$}}
\end{aligned}
\]

\[
(1)\qquad \underline{\Delta}_H(E,q) \xrightarrow{\ p_{H',H}\ } \underline{\Delta}_{H'}(E,q)
\]

\[
\underline{\mathrm{Arf}}(E,q) = \text{schéma des comp. conn. de } \underline{\Delta}_{\{n\}}(E,q) \text{ sur } S
\]

\[
\underline{\Delta}_H \longrightarrow \underline{\Delta}_{\{n\}} \longrightarrow \underline{\mathrm{Arf}}
\]

\[
(2)\qquad \underline{\Delta}_H \longrightarrow \underline{\Delta}_{H'}\times_S \underline{\mathrm{Arf}}
\]

\[
(3)\qquad \underline{\Delta}_{\{n,n-1\}} \longrightarrow \underline{\Delta}_{\{n-1\}}
\]

\[
\underline{\Delta}_{\{n,n-1\}} \xrightarrow{\ \sim\ } \underline{\Delta}_{\{n-1\}}\times \underline{\mathrm{Arf}}
\qquad (\text{pour } N=2n).
\]

\[
(4)\qquad \underline{\mathrm{Arf}}' = \underline{M}\bigl((-1)^n\delta(q)^{-1}\bigr),
\]

\[
(5)\qquad \underline{\mathrm{Arf}} \longrightarrow \underline{\mathrm{Arf}}'
\]

\[
(6)\qquad
\begin{cases}
\underline{\Delta}_{\{n\}} \longrightarrow \underline{\mathrm{Arf}}' \\
F \longmapsto \omega_F \in \underline{\det}(E)^{*} \simeq \underline{\mathrm{Hom}}\bigl(\underbrace{\underline{\det} F\otimes\underline{\det}\check F}_{\underline{O}_S},\ \underline{\det} E\bigr)
\end{cases}
\]

\[
(7)\qquad \underline{\mathrm{Arf}}' \simeq \underline{\mathrm{Arf}} \overset{(\mathbb{Z}/2)_S}{\wedge} \mu_{2,S}
\]

\[
\underline{\mathrm{Arf}} \xrightarrow{\ \sim\ } \underline{\mathrm{Arf}}' = \underline{M}\bigl((-1)^n\delta(q^{*})^{-1}\bigr)
\]

\[
\underline{O}(q) \simeq \mu_2\times\underline{SO}(q) \qquad (\mu_2 = \text{\uncertain{centre}})
\]

\[
\begin{cases}
\underline{SO}(q) \longrightarrow \underline{\Delta}_{\{n\}} \\
u \longmapsto u(F)
\end{cases}
\]

\[
\begin{cases}
\underline{SO}(q) \longrightarrow \underline{\Delta}_H \\
u \longmapsto u(D)
\end{cases}
\]

\[
e_i\mapsto e'_i,\qquad e_i^{*}\mapsto e'^{*}_i ,
\]

\[
E \xrightarrow[\text{défini par }\varphi]{\ \psi\ } E^{\vee} \xrightarrow{\ \text{restr}\ } F^{\vee}
\]

\[
F\times G \longrightarrow \underline{O}_S
\]

\[
G \xrightarrow{\ \sim\ } \check F,\qquad F \xrightarrow{\ \sim\ } \check G .
\]

\[
0\longrightarrow F^{\perp} \longrightarrow E \longrightarrow G \longrightarrow 0,
\qquad F\subset F^{\perp}
\]

\[
\underline{\mathrm{Splitisot}}(E,F^{\perp}) \hookrightarrow \underline{\mathrm{Split}}(E,F^{\perp}).
\]

\[
0\longrightarrow \underline{\mathrm{Hom}}(G,F) \longrightarrow \underline{\mathrm{Hom}}(G,F^{\perp}) \longrightarrow \underline{\mathrm{Hom}}(G,Q) \longrightarrow 0
\]

\[
Q = F^{\perp}/F .
\]

\[
\theta : G \longrightarrow E
\]

\[
(*)\qquad 0\longrightarrow F^{\perp} \longrightarrow E \longrightarrow G \longrightarrow 0
\]

\[
q_\theta : F\times G \longrightarrow \underline{O}_S
\]

\[
q_\theta(x,y) = q\bigl(x+\theta(y)\bigr) = q(\theta(y)) + \langle y,x\rangle .
\]

\[
\begin{aligned}
\varphi_\theta(x,y;x',y') &= \varphi(x+\theta y,\theta y') + \varphi(x',\theta y)\\
&= \varphi(\theta y,\theta y') + \langle y',x\rangle + \langle y,x'\rangle
\end{aligned}
\]

\[
u : G \longrightarrow F
\]

\[
\theta_u = \theta + u
\]

\[
q\circ\theta_u = 0
\]

\[
u \in \mathrm{Hom}(G,F) \simeq \mathrm{Bil}(G,G) \qquad (F\simeq\check G)
\]

\[
\mathrm{Alt}(G,G) \hookrightarrow \mathrm{Bil}(G,G) \simeq \mathrm{Hom}(G,F) \hookrightarrow \mathrm{Hom}(G,F^{\perp})
\]

\[
q(x,x') = \langle x,x'\rangle ,
\]

\[
D :\ 0 \subsetneq E_1 \subsetneq E_2 \subset\cdots\subsetneq E_\nu = F \qquad E ,
\]

\[
0\subsetneq E_1 \subset\cdots\subsetneq E_\nu = F \subset F^{\perp} \subsetneq E_{\nu-1}^{\perp} \subsetneq\cdots\subsetneq E_1^{\perp} \subsetneq F
\]

\[
D\in\Delta_H(E,q),\qquad D'\in\Delta_H(E',q') ,
\]

\[
\exists\, u : (E,q) \xrightarrow{\ \sim\ } (E',q') \quad\text{tel que}\quad u(D)=D' .
\]

\[
u : (E,q) \longrightarrow (E',q') \quad\text{tel que}\quad u(D)=D' .
\]

\[
\begin{cases}
\underline{O}(q) \longrightarrow \underline{\Delta}_H \\
u \longmapsto u(D)
\end{cases}
\]

\[
\underline{\Delta}_H \xleftarrow{\ \sim\ } \underline{O}(q)/\underline{\mathrm{Stab}}_D(q)
\]

\[
R_i = E_i/E_{i-1},\qquad R'_i = E'_i/E'_{i-1}\qquad (E_0=0,\ E'_0=0)
\]

\[
\begin{aligned}
u_i &: R_i \xrightarrow{\ \sim\ } R'_i\\
u_q &: Q \xrightarrow{\ \sim\ } Q'
\end{aligned}
\]

\[
u : (E,q) \longrightarrow (E',q')
\]

\[
\overline{\Sigma} = \prod_{1\le i\le\nu} \Sigma_i \times \Sigma_0
\]

\[
0\longrightarrow E_{i-1} \longrightarrow E_i \longrightarrow R_i \longrightarrow 0
\]

\[
0\longrightarrow E_\nu \longrightarrow E_\nu^{\perp} \longrightarrow Q \longrightarrow 0 .
\]

\[
\overline{E} = \bigoplus_i \bigl(R_i + \check R_i\bigr) \oplus Q
\]

\[
\overline{E}_i = \sum_{1\le j\le i} R_j \subset \overline{E} .
\]

\[
\begin{cases}
\overline{R}_i = \overline{E}_i/\overline{E}_{i-1}\\
\overline{Q} = \overline{E}_\nu^{\perp}/\overline{E}_\nu
\end{cases}
\]

\[
\begin{aligned}
\overline{R}_i &\xrightarrow[\ \overline{u}_i\ ]{\ \sim\ } R_i\\
\overline{Q} &\xrightarrow[\ \overline{u}_q\ ]{\ \sim\ } Q ,
\end{aligned}
\]

\[
\underline{\mathrm{Isom}}_{\{\overline{u}_i\},\overline{u}}\bigl((\overline{E},\overline{q},\overline{D});(E,q,D)\bigr)
\]

\[
\check{\mathbb{V}}\Bigl(\sum_{1\le i\le\nu}\underline{\mathrm{Hom}}(R_i,E_{i-1})\oplus\underline{\mathrm{Hom}}(\overline{G},\overline{F}^{\perp})^{!}\Bigr)
\]

\[
\begin{cases}
\underline{\mathrm{Hom}}(\overline{G},\overline{F}^{\perp}) \longrightarrow \underline{\mathrm{Quad}}(\overline{G})\\
u \longmapsto \bigl(\overline{y}\mapsto \overline{q}(\overline{y}+u(\overline{y}))\bigr)
\end{cases}
\]

\[
\operatorname{rg} R_i = d_i,\qquad \operatorname{rg} Q = \rho \qquad\text{donc}\quad N = 2\overbrace{\textstyle\sum d_i}^{d}+\rho,
\]

\[
\operatorname{rg}\overline{F} = \operatorname{rg} G = \sum d_i \overset{\text{déf}}{=} d
\]

\[
\begin{cases}
\dim_S \underline{\Sigma}_i = d_i \displaystyle\sum_{1\le j\le i-1} d_j\\[1ex]
\dim_S \underline{\Sigma}_0 = \dim_S \underline{\mathrm{Hom}}(\overline{G},\overline{F}^{\perp}) - \dim_S \underline{\mathrm{Quad}}(\overline{G})\\[1ex]
\phantom{\dim_S \underline{\Sigma}_0} = d(d+\rho) - \dfrac{d(d+1)}{2} = \dfrac{d(d-1)}{2}+d\rho
\end{cases}
\]

\[
\begin{aligned}
\dim_S \underline{O}_D(q)^{+} &= \underbrace{\sum_{1\le j<i\le\nu} d_i d_j}_{\frac12(d^2-\sum d_i^2)} + \frac{d(d-1)}{2} + d\rho\\
&= d^2 + d\rho - \sum\frac{d_i(d_i+1)}{2}
\end{aligned}
\]

\[
\dim_S\Bigl(\prod \mathrm{Gl}(R_i)\times\underline{O}(q_Q)\Bigr) = \sum d_i^2 + \frac{\rho(\rho-1)}{2}
\]

\[
\dim_S \underline{O}_D(q) = d^2 + d\rho + \frac{\rho(\rho-1)}{2} + \sum_i \frac{d_i(d_i-1)}{2}
\]

\[
\dim_S \underline{\Delta}_D(q) = \underbrace{\dim_S \underline{O}(q)}_{\frac{N(N-1)}{2} = 2d^2+d(2\rho-1)+\frac{\rho(\rho-1)}{2}} - \dim_S \underline{O}_D(q) = \text{\struck{$\frac{N(N-1)}{2}$}}
\]

\[
\dim_S \underline{\Delta}_H(q) = d^2 + d\rho - d - \sum\frac{d_i(d_i-1)}{2}
\]

\[
\dim_S \underline{\Delta}_{\{d\}}(q) = \frac{d(d+1)}{2} + d(\rho-1) \qquad (\rho = N-2d)
\]

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

\[
\underline{O}(q) = \underline{SO}(q)\times\mu_2 ,\qquad
\text{\struck{où}}\ 
\begin{cases}
\mu_2 = \underline{O}(q)\cap\mathbb{G}_m = Z(\underline{O}(q))\\
\underline{SO}(q) = \underline{O}(q)\cap\underline{Sl}(E)
\end{cases}
\]

\[
\underline{SO}_D(q) = \underline{O}_D(q)\cap\underline{Sl}(E)
\]

\[
\underline{O}_D(q) \xrightarrow{\ \alpha\ } \prod_{1\le i\le\nu}\underline{Gl}(R_i)\times\underline{O}(q_Q)
\]

\[
\underline{SO}_D(q) = \alpha^{-1}\bigl(\underline{SO}(q_Q)\bigr) .
\]

\[
\underline{O}(q) \xrightarrow{\ \varepsilon_q\ } (\mathbb{Z}/2)_S
\]

\[
\underline{SO}(q) = \operatorname{Ker}\varepsilon
\]

\[
1\longrightarrow \underline{SO}(q) \longrightarrow \underline{O}(q) \xrightarrow{\ \varepsilon_q\ } (\mathbb{Z}/2)_S \longrightarrow 1
\]

\[
1\longrightarrow \underline{O}_D(q)^{+} \longrightarrow \underline{O}_D(q) \longrightarrow \prod_{1\le i\le\nu}\underline{Gl}(R_i)\times\underline{O}(q_Q) \longrightarrow 1
\]

\[
\underline{O}_D(q) \longrightarrow \underline{O}(q_Q) \xrightarrow{\ \varepsilon_{q_Q}\ } (\mathbb{Z}/2)_S ,
\]