Page 63 - Geometric Modeling and Algebraic Geometry

60 P. H. Johansen et al.
Recall that an A n singularity is a singularity with normal form x + x + x n+1 ,
2
2
1 2
3
see [3, p. 184].
Proposition 6. Let f d−1 and f d be as in Definition 2, and assume P =(p 0 : p 1 :
p 2 : p 3 ) =(1:0:0:0) is a singular point of Z(F) with I (p 1 :p 2 :p 3 ) (f d−1 ,f d )= m.
Then P is an A m−1 singularity.
Proof. We may assume that P =(0:0:0:1) and write the local equation
d

g := F(x 0 ,x 1 ,x 2 , 1) = x 0 f d−1 (x 1 ,x 2 , 1) + f d (x 1 ,x 2 , 1) = g i (4.2)
i=2
¯
with g i ∈ k[x 0 ,x 1 ,x 2 ] homogeneous of degree i. Since Z(f d−1 ) is nonsingular at
0:=(0:0:1), we can assume that the linear term of f d−1 (x 1 ,x 2 , 1) is equal to
x 1 . The quadratic term g 2 of g is then g 2 = x 0 x 1 + ax + bx 1 x 2 + cx for some
2
2
1 2
a, b, c ∈ k. The Hessian matrix of g evaluated at P is
⎛ ⎞
01 0
H(g)(0, 0, 0) = H(g 2 )(0, 0, 0) = ⎝ 12ab ⎠
0 b 2c

which has corank 0 when c =0 and corank 1 when c =0. By [3, p. 188], P is an A 1
singularity when c =0 and an A n singularity for some n when c =0.
The index n of the singularity is equal to the Milnor number
¯ ¯
.
k[x 0 ,x 1 ,x 2 ] (x 0 ,x 1 ,x 2 )
k[x 0 ,x 1 ,x 2 ] (x 0 ,x 1 ,x 2 )
µ =dim¯ k =dim¯ k
, ∂x 1 , ∂x 2
J g ∂g ∂g ∂g
∂x 0
We need to show that µ =I 0 (f d−1 ,f d ) − 1. From the definition of the intersection
multiplicity, it is not hard to see that
¯
.
k[x 1 ,x 2 ] (x 1 ,x 2 )
I 0 (f d−1 ,f d )=dim¯ k
(f d−1 (x 1 ,x 2 , 1),f d (x 1 ,x 2 , 1))
The singularity at p is isolated, so the Milnor number is finite. Furthermore, since
gcd(f d−1 ,f d )=1, the intersection multiplicity is finite. Therefore both dimensions
can be calculated in the completion rings. For the rest of the proof we view f d−1
¯
¯
and f d as elements of the power series rings k[[x 1 ,x 2 ]] ⊂ k[[x 0 ,x 1 ,x 2 ]], and all
calculations are done in these rings.
Since Z(f d−1 ) is smooth at O, we can write
f d−1 (x 1 ,x 2 , 1) = (x 1 − ϕ(x 2 )) u(x 1 ,x 2 )

for some power series ϕ(x 2 ) and invertible power series u(x 1 ,x 2 ). To simplify no-
¯
tation we write u = u(x 1 ,x 2 ) ∈ k[[x 1 ,x 2 ]].
The Jacobian ideal J g is generated by the three partial derivatives: