Page 64 - Geometric Modeling and Algebraic Geometry

4 Monoid Hypersurfaces 61
∂g
=(x 1 − ϕ(x 2 )) u
∂x 0

∂g ∂u ∂f d
= x 0 u +(x 1 − ϕ(x 2 )) + (x 1 ,x 2 )
∂x 1

∂x 1 ∂x 1
∂g ∂u ∂f d

= x 0 −ϕ (x 2 )u +(x 1 − ϕ(x 2 )) ∂x 2 + ∂x 2 (x 1 ,x 2 )
∂x 2

∂g
By using the fact that x 1 − ϕ(x 2 ) ∈ we can write J g without the symbols
∂u ∂u ∂x 0
and ∂x 2 :
∂x 1

J g = x 1 − ϕ(x 2 ),x 0 u + ∂f d (x 1 ,x 2 ), −x 0 uϕ (x 2 )+ ∂x 2 (x 1 ,x 2 )

∂f d
∂x 1
To make the following calculations clear, define the polynomials h i by writing
d i
f d (x 1 ,x 2 , 1) = x h i (x 2 ).Now
i=0 1

d d i

J g = x 1 − ϕ(x 2 ),x 0 u + ix i−1 h i (x 2 ), −x 0 uϕ (x 2 )+ x h (x 2 ) ,
1 i
i=1 1 i=0
so
¯ ¯
k[[x 2 ]]
k[[x 0 ,x 1 ,x 2 ]]
=
J g (A(x 2 ))
where

d d
A(x 2 )= ϕ (x 2 ) iϕ(x 2 ) i−1 h i (x 2 ) + ϕ(x 2 ) h (x 2 ) .
i

i
i=1 i=0
For the intersection multiplicity we have
¯
¯ k[[x 1 ,x 2 ]] ¯
k[[x 1 ,x 2 ]]
k[[x 2 ]]
= =
d i
f d−1 (x 1 ,x 2 , 1),f d (x 1 ,x 2 , 1) x 1 − ϕ(x 2 ), x h i (x 2 ) B(x 2 )
i=0 1
d i
where B(x 2 )= ϕ(x 2 ) h i (x 2 ). Observing that B (x 2 )= A(x 2 ) gives the

i=0
result µ =I 0 (f d−1 ,f d ) − 1.
Corollary 7. A monoid surface of degree d can have at most d(d − 1) singularities
1
2
in addition to O. If this number of singularities is obtained, then all of them will be
of type A 1 .
Proof. The sum of all local intersection numbers I a (f d−1 ,f d ) is givenbyB´ ezout’s
theorem:

I a (f d−1 ,f d )= d(d − 1).
a∈Z(f d−1 ,f d )
The line L a will contain a singularity other than O only if I a (f d−1 ,f d ) ≥ 2, giving a
maximum of d(d−1) singularities in addition to O. Also, if this number is obtained,
1
2
all local intersection numbers must be exactly 2, so all singularities other than O will
be of type A 1 .