Page 72 - Geometric Modeling and Algebraic Geometry

4 Monoid Hypersurfaces 69
When m =0, one condition must be satisfied for q to be of the form f 4 (θ),
namely b 11 =0, where b 11 is the coefficient of st 11 in q.
For example, we can have an A 11 singularity only if q is of the form (αs−βt) .
12
The condition b 11 =0 implies that either q = λs 12 or q = λt . The first case gives
12
a surface with a singular line, while the other gives a monoid with an A 11 singularity
(see Figure 4.2). The line from O to the A 11 singularity corresponds to the inflection
point of Z(f 3 ).
For any set of multiplicities m 1 ,...,m r with m 1 +···+m r =12, it is not hard to
see that there exist real points p 1 ,...,p r such that the condition b 11 =0 is satisfied.

It suffices to take p i =(α i :1), with m i α i =0 (the condition corresponding to
b 11 =0). This completely classifies the possible configurations of singularities when
f 3 is a cuspidal curve.
Case 3. The tangent cone is the product of a conic and a line that is not tangent
to the conic, and we can assume f 3 = x 3 (x 1 x 2 + x ). Then Z(f 3 ) is singular at
2
3
(1 :0:0) and (0 :1:0), the intersections of the conic Z(x 1 x 2 + x ) and the line
2
3
Z(x 3 ). For each f 4 we can associate four integers:
j 0 := I (1:0:0) (x 1 x 2 + x ,f 4 ), k 0 := I (1:0:0) (x 3 ,f 4 ),
2
3
j 1 := I (0:1:0) (x 1 x 2 + x ,f 4 ), k 1 := I (0:1:0) (x 3 ,f 4 ).
2
3
We see that k 0 > 0 ⇔ f 4 (1 :0:0)=0 ⇔ j 0 > 0, and that Z(f 4 ) is singular
at (1 :0:0) if and only if k 0 and j 0 both are bigger than one. These cases imply
a singular line on the monoid, and are not considered in this article. The same holds
for k 1 , j 1 and the point (0 :1:0).
Define r i = max(j i ,k i ) for i =1, 2. Then, by [19], O will be a singularity of
if r 0 ≥ r 1 .
if r 0 ≤ r 1 ,oroftype T 3,4+r 1 ,4+r 0
type T 3,4+r 0 ,4+r 1
We can parameterize the line Z(x 3 ) by θ 1 where θ 1 (s, t)=(s, t, 0), and the conic
Z(x 1 x 2 + x ) by θ 2 where θ 2 (s, t)=(s , −t ,st). Similarly to the previous cases,
2
2
2
3
roots of f 4 (θ 1 ) correspond to intersections between Z(f 4 ) and the line Z(x 3 ), while
roots of f 4 (θ 2 ) correspond to intersections between Z(f 4 ) and the conic Z(x 1 x 3 +
x ).
2
3
For any legal values of of j 0 , j 1 , k 0 and k 1 , parameter points
) ∈ P \{(0 : 1), (1 : 0)},
1
(α 1 : β 1 ),..., (α m r : β m r
with multiplicities m 1 ,...,m r such that m 1 +···+m r =4−k 0 −k 1 , and parameter
points

(α : β ),..., (α m : β m ) ∈ P \{(0 : 1), (1 : 0)},

1
1 1 r r

with multiplicities m ,...,m such that m + ··· + m =8 − j 0 − j 1 , we can fix

r
r
1 1
polynomials q 1 and q 2 such that
• q 1 is nonzero, of degree 4, and has factors s , t k 0 and (β i s − α i t) m i for i =
k 1
1,...,r,

• q 2 is nonzero, of degree 8, and has factors s , t j 0 and (β s − α t) m i for i =
j 1
i i
1,...,r .