Introduction - Laboratoire Paul Painlevé

Q
Q
Q
Q
1
xy +zw +z3+w3= 0
P2
k
X Y X ×kY
Q
k
Q
X Y Q k
X×kY
Q
Q
Q

Q
Q
Q
XCl(X)
XCa(X)X
Ca(X)→Cl(X)
X
Ca(X)→Cl(X)
Q
QCl(OX,x)
xCl(X)/Ca(X)
Pic(X)X
Ca(X)X U ⊂X
2
Cl(X)→Cl(U)
X U X
Pic(X)→Pic(U)
FX
F F → F∨∨
FV⊂X Y ⊂V
F(V)→ F(V\Y)
F
X
0→ F → E → G → 0
E G
F
F
F
FW⊂X Y ⊂W
FW∼
=j∗FW\Yj:W\Y →W

X j :V →X
codimX(X\V)≥2EV
j∗EX
j∗E
codimX(X\V)≥2j∗E
j∗E(W)→j∗E(W\Y)
E(W∩V)→ E((W\Y)∩V) = E((W∩V)\(Y∩V)) Y∩V
VE
X i:U →X
D
OX(D) = i∗OU(D|U)
U D
j:V →X V U E V D
E X
OX(D)∼
=j∗OV(E)U
XOX(D)
D
S X S F
X s S x
X k(s)FsXs
XsxFx
Fs1x M
OX,x FxmsS s OXs,x (Fs)x
(Fs)x=M⊗OS,s k(s)∼
=M/msM1g
M M/msM M =gOX,x +msM
msOX,x mxOX,x
M g
1
X Y
F1X×kY x X
F|{x}×YYF
{x} × Y
Xk
P(X) Cla(X)
X
Pic0(X)X
Albm(X)X X
Alb(X)
X
P(X)
ak
X αX:X99K Alb(X)

Xk
a0VX
UXX
P(X) = Alb(X)∨
P(X)k
P0
XSL
UX×SL|{a}×SL|UX×{s}
ks S
PAlb(X)×P(X)E
UX×P(X) P(X)P0
X
P(X)
P0
Xk
α: P(X)−→ P0
X
E4 5 6 7
k
α(k)
αXAlb(X)
Hk k
β:H→P0
X
H0H
A L L
β L
H
H0H U ×H0
Alb(X)→(H0)∨ϕ0:H0→P(X)
β|H0=α◦ϕ0H→H/H0
P0
X(k) P(X)
ϕ:H→P(X)β=α◦ϕ
4 5
6Tkξ:T→P0
X
LU×T
T ν :e T→Te L
LU×e T N(ξ)ν
e
N(ξ) = {˜ t∈e T|e L|U×{˜
t}' OU×{˜ t}} ⊂ e
T U
X2e N(ξ)
X×e T→e Te Le L−1
f:T0→T
N(ξ)⊂T ξ ◦f= 0 T N(ξ)
7CkT⊂C
ξ:C\T→P0
X
OX(1) X d X
m1. . . . , md−1
h1, . . . , hd−1OX(m1),...,OX(md−1)
Y U
Alb(Y)→Alb(U)

ρ:P0
X(k)→P0 Y(k)P0 Y
Pic0(Y)Y
C\T(k)→P0
Y(k)ξ
mTm
ξ
X×Pic0(X)red
Pic0(X)red →P(X)
Alb(X)→Albm(X)
X→Albm(X)
X99K Alb(X)
X
k
Q
Alb(X)→Albm(X)
P(X)/Pic0(X)red
XQ
X→Alb(X)X
Q
G
g>2G
Q
XNS(X) = Cl(X)/Cla(X)
X X
kNS(X)
k
Q
m
X
NS(X)
m m 
Xfact X
x∈XOX,x
X U
X
U