Étude des courbes algébriques planes

C2
P2(C)
P2(C)
deg C ≥2
C∗
P2(C)
n∗=n(n−1) −2d−3s
s∗= 3n(n−2) −6d−8s
n=n∗(n∗−1) −2d∗−3s∗
s= 3n∗(n∗−2) −6d∗−8s∗
d s
d∗C∗s∗
C∗
C2
f∈C[X1, X2]
C=V(f) = {(x1, x2)∈C2|f(x1, x2) = 0}

C1C2C16=C2C=C1∪C2
C1C2
C=C1∪C2C1=C2
C[X1, X2]
{fi}i∈[[1,...,k]] f=f1r1...fkrkri6= 0 C[X1, X2]
C
˜
f=f1. . . fk
˜
f
˜
f
deg C =deg ˜ f
˜
f
C2
P2(C)
P(X1, . . . , Xk) = X
i1+···+ik=n
αi1,...,ikXi1
1. . . Xik
k
αi1,...,ik∈C
f(X1, X2) = f0(X1, X2) + · · · +fn(X1, X2)
fi
F(X0, X1, X2) = Xn
0f0(X1, X2) + · · · +X0 0fn(X1, X2)
F=Xn
0f(X1
X0
,X2
X0
)et f =F(1, X1, X2)

¯
C=V(F)⊂P2(C)
P2(C)C C3\ {0}
(x0, x1, x2) = λ(y0, y1, y2)⇔(x0, x1, x2)∼(y0, y1, y2)
(x0:x1:x2)¯ C
P2(C)
F∈C[X0, X1, X2]
K=V(F)et degF ≥1
C2P2(C)
P(X) = p0+p1X+· · · +pnXn
Q(X) = q0+q1X+· · · +qmXmA[X]
piqiRP,Q
RP,Q =
p0. . . pn
p0. . . pn
q0. . . . . . qm
q0. . . . . . qm
=det(MP,Q)
MP,Q m+n
C1=V(F1)C2=V(F2)
P2(C)
q= (0 : 0 : 1)
F1(0 : 0 : 1) 6= 0 et F2(0 : 0 : 1) 6= 0
p= (p0:p1:p2)∈C1∩C2
multp(C1∩C2) = ordp0(G)G∈C[X0, X1]F1F2
p0= (p0:p1)ordp0(G)

f(X1, X2) = X1(X1+ 1) + X2
2= 0
X1= 0 C2
F(X0, X1, X2) = X1(X1+X0) + X2
2
(X1(X0+X1) + X2
2= 0
X1= 0
X1= 0
(X2
2= 0
X1= 0
p= (1 : 0 : 0)
X1= 0
A[X1]
A=C[X0, X2]
G(X0, X2) = 
1 0 0
0 1 0
1X0X2
2
=X2
2
˜p= (1 : 0)
p= (1 : : 0)
X2
X1
X1= 0