Page 75 - Geometric Modeling and Algebraic Geometry

72 P. H. Johansen et al.
The three lines Z(x 1 ), Z(x 2 ) and Z(x 3 ) are parameterized by θ 1 , θ 2 and θ 3 where
θ 1 (s, t)=(0,s,t), θ 2 (s, t)=(s, 0,t) and θ 3 (s, t)=(s, t, 0). Roots of the polyno-
mial f 4 (θ i ) away from (1 : 0) and (0 : 1) correspond to intersections between Z(f 4 )
and Z(x i ) away from the singular points of Z(f 3 ).
As before, we are only interested in the cases where none of f 4 (θ i ) ≡ 0 for
i =1, 2, 3, as this would make the monoid reducible.
For the study of other singularities on the monoid we consider nonzero polyno-
mials
q 1 = b 0 s + b 1 s t + b 2 s t + b 3 st + b 4 t ,
2 2
3
4
3
4
q 2 = c 0 s + c 1 s t + c 2 s t + c 3 st + c 4 t ,
4
3
3
4
2 2
q 3 = d 0 s + d 1 s t + d 2 s t + d 3 st + d 4 t .
2 2
4
4
3
3
Linear algebra shows that (λ 1 q 1 ,λ 2 q 2 ,λ 3 q 3 )=(f 4 (θ 1 ),f 4 (θ 2 ),f 4 (θ 3 )) for some
f 4 if and only if λ 1 b 0 = λ 3 d 4 , λ 1 b 4 = λ 2 c 4 , and λ 2 c 0 = λ 3 d 0 . A simple analysis
shows the following: There exist λ 1 ,λ 2 ,λ 3 =0 such that
(λ 1 q 1 ,λ 2 q 2 ,λ 3 q 3 )=(f 4 (θ 1 ),f 4 (θ 2 ),f 4 (θ 3 ))
for some f 4 , and such that Z(f 4 ) and Z(f 3 ) have no common singular point if and
only if all of the following hold:
• b 0 =0 ↔ d 4 =0 and b 0 = d 4 =0 → (b 1 =0 or d 3 =0),
• b 4 =0 ↔ c 4 =0 and b 4 = c 4 =0 → (b 3 =0 or c 3 =0),
• c 0 =4 ↔ d 0 =0 and c 0 = d 0 =0 → (c 1 =0 or d 1 =0),
• b 0 c 4 d 0 = b 4 c 0 d 4 .
Similarly to the previous cases we can classify the possible configurations of
other singularities by varying the multiplicities of the roots of the polynomials q 1 , q 2
and q 3 . Only the multiplicities of the roots (0 : 1) and (1 : 0) affect the first three
bullet points above. Then, for any set of multiplicities of the rest of the roots, we
can find q 1 , q 2 and q 3 such that the last bullet point is satisfied. This completes the
classification when Z(f 3 ) is the product of three general lines.
Case 6. The tangent cone is three lines meeting in a point, and we can assume
that f 3 = x − x 2 x . We write f 3 =
1
2
3 where
1 = x 2 ,
2 = x 2 − x 3 and
3
2
2 3

3 = x 2 +x 3 , representing the three lines going through the singular point (1 :0:0).
For each f 4 we associate three integers j 1 , j 2 and j 3 defined as the intersection
numbers j i =I (1:0:0) (f 4 ,
i ). We see that j 1 =0 ⇔ j 2 =0 ⇔ j 3 =0, and that
Z(f 4 ) is singular at (1 :0:0) if and only if two of the integers j 1 , j 2 , j 3 are greater
then one. (Then all of them will be greater than one.) The singularity will be of the
U series [1], [2].
The three lines Z(
1 ), Z(
2 ) and Z(
3 ) can be parameterized by θ 1 , θ 2 , and θ 3
where θ 1 (s, t)=(s, 0,t), θ 2 (s, t)=(s, t, t) and θ 2 (s, t)=(s, t, −t).
For the study of other singularities on the monoid we consider nonzero polyno-
mials