Page 73 - Geometric Modeling and Algebraic Geometry

70 P. H. Johansen et al.
Now q 1 and q 2 are determined up to multiplication by nonzero constants. Write q 1 =
b 0 s + ··· + b 4 t and q 2 = c 0 s + ··· + c 8 t .
4
4
8
8
The classification of singularities on the monoid consists of describing the con-
ditions on the parameter points and nonzero constants λ 1 and λ 2 for the pair
(λ 1 q 1 ,λ 2 q 2 ) to be on the form (f 4 (θ 1 ),f 4 (θ 2 )) for some f 4 .
Similarly to the previous cases, f 4 (θ 1 ) ≡ 0 if and only if x 3 is a factor in f 4 and
f 4 (θ 2 ) ≡ 0 if and only if x 1 x 2 + x is a factor in f 4 . Since f 3 = x 3 (x 1 x 2 + x ),
2
2
3 3
both cases will make the monoid reducible, so we only consider λ 1 ,λ 2 =0.
We use linear algebra to study the relationship between the coefficients a 1 ...a 15
of f 4 and the polynomials q 1 and q 2 .Wefind (λ 1 q 1 ,λ 2 q 2 ) to be of the form
(f 4 (θ 1 ),f 4 (θ 2 )) if and only if λ 1 b 0 = λ 2 c 0 and λ 1 b 4 = λ 2 c 8 . Furthermore, the
pair (λ 1 q 1 ,λ 2 q 2 ) will fix f 4 modulo f 3 . Since f 4 and λf 4 correspond to projectively
equivalent monoids for any λ =0, it is the ratio λ 1 /λ 2 , and not λ 1 and λ 2 , that is
important.
Recall that k 0 > 0 ⇔ j 0 > 0 and k 1 > 0 ⇔ j 1 > 0.If k 0 > 0 and k 1 > 0,
then b 0 = c 0 = b 4 = c 8 =0, so for any λ 1 ,λ 2 =0 we have (λ 1 q 1 ,λ 2 q 2 )=
(f 4 (θ 1 ),f 4 (θ 2 )) for some f 4 . Varying λ 1 /λ 2 will give a one-parameter family of
monoids for each choice of multiplicities and parameter points.
If k 0 =0 and k 1 > 0, then b 0 = c 0 =0. The condition λ 1 b 4 = λ 2 c 8 implies
λ 1 /λ 2 = c 8 /b 4 . This means that any choice of multiplicities and parameter points
will give a unique monoid up to projective equivalence. The same goes for the case
where k 0 > 0 and k 1 =0.
Finally, consider the case where k 0 = k 1 =0.For (λ 1 q 1 ,λ 2 q 2 ) to be of the
form (f 4 (θ 1 ),f 4 (θ 2 )) we must have λ 1 /λ 2 = c 8 /b 4 = c 0 /b 0 . This translates into a
condition on the parameter points, namely

m 1 ··· (α )
(β ) m r (α ) m
m 1 ··· (β )
r
r r
1 = 1 . (4.4)
β m 1 ··· β r m r α m 1 ··· α r m r
1 1
In other words, if condition (4.4) holds, we have a unique monoid up to projective
equivalence.
It is easy to see that for any choice of multiplicities, it is possible to find real pa-
rameter points such that condition (4.4) is satisfied. This completes the classification
of possible singularities when the tangent cone is a conic plus a chordal line.
Case 4. The tangent cone is the product of a conic and a line tangent to the conic,
and we can assume f 3 = x 3 (x 1 x 3 + x ).Now Z(f 3 ) is singular at (1 :0:0).For
2
2
each f 4 we can associate two integers
j 0 := I (1:0:0) (x 1 x 3 + x ,f 4 ) and k 0 := I (1:0:0) (x 3 ,f 4 ).
2
2
We have j 0 > 0 ⇔ k 0 > 0, j 0 > 1 ⇔ k 0 > 1. Furthermore, j 0 and k 0 are both
greater than 2 if and only if Z(f 4 ) is singular at (1 :0:0), a case we have excluded.
The singularity at O will be of the S series, from [1], [2].
2
We can parameterize the conic Z(x 1 x 3 + x ) by θ 2 and the line Z(x 3 ) by θ 1
2
where θ 2 (s, t)=(s ,st, −t ) and θ 1 (s, t)=(s, t, 0). As in the previous case, the
2
2