Page 76 - Geometric Modeling and Algebraic Geometry

4 Monoid Hypersurfaces 73
q 1 = b 0 s + b 1 s t + b 2 s t + b 3 st + b 4 t ,
4
3
3
2 2
4
q 2 = c 0 s + c 1 s t + c 2 s t + c 3 st + c 4 t ,
2 2
4
3
4
3
q 3 = d 0 s + d 1 s t + d 2 s t + d 3 st + d 4 t .
4
3
2 2
4
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 2 c 4 = λ 3 d 0 , and 2λ 1 b 1 = λ 2 c 1 + λ 3 d 1 . 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 ↔ c 0 =0 ↔ d 0 =0,
• if b 0 = c 0 = d 0 =0, then at least two of b 1 , c 1 , and d 1 are different from zero,
• 2b 1 c 0 d 0 = b 0 c 1 d 0 + b 0 c 0 d 1 .
As in all the previous cases we can classify the possible configurations of other
singularities for all possible j 1 ,j 2 ,j 3 . As before, the first bullet point only affect the
multiplicity of the factor t in q 1 , q 2 and q 3 . For any set of multiplicities for the rest
of the roots, we can find q 1 , q 2 , q 3 with real roots of the given multiplicities such that
the last bullet point is satisfied. This completes the classification of the singularities
(other than O) when Z(f 3 ) is three lines meeting in a point.
Case 7. The tangent cone is a double line plus a line, and we can assume
f 3 = x 2 x . The tangent cone is singular along the line Z(x 3 ). The line Z(x 2 ) is para-
2
3
meterized by θ 1 and the line Z(x 3 ) is parameterized by θ 2 where θ 1 (s, t)=(s, 0,t)
and θ 2 (s, t)=(s, t, 0). The monoid is reducible if and only if f 4 (θ 1 ) or f 4 (θ 2 ) is
identically zero, so we assume that neither is identically zero. For each f 4 we asso-
ciate two integers, j 0 := I (1:0:0) (f 4 ,x 2 ) and k 0 := I (1:0:0) (f 4 ,x 3 ). Furthermore, we
write f 4 (θ 2 ) as a product of linear factors
r

f 4 (θ 2 )= λs k 0 (α i s − t) m i .
i=0
Now the singularity at O will be of the V series and depends on j 0 , k 0 and
m 1 ,...,m r .
Other singularities on the monoid correspond to intersections of Z(f 4 ) and the
line Z(x 2 ) away from (1 :0:0). Each such intersection corresponds to a root in the
polynomial f 4 (θ 1 ) different from (1 : 0). Let j ≤ 4 and k ≤ 4 be integers such that

0 0
j 0 > 0 ↔ k 0 > 0. Then, for any homogeneous polynomials q 1 , q 2 in s, t of degree
4 such that s is a factor of multiplicity j in q 1 and of multiplicity k in q 2 , there is

0 0

a polynomial f 4 and nonzero constants λ 1 and λ 2 such that k 0 = k , j 0 = j and

0 0
(λ 1 q 1 ,λ 2 q 2 )=(f 4 (θ 1 ),f 4 (θ 2 )). Furthermore, if q 1 and q 2 have real coefficients,
then f 4 can be selected with real coefficients. This follows from an analysis similar
to case 5 and completes the classification of singularities when the tangent cone is a
product of a line and a double line.
Case 8. The tangent cone is a triple line, and we assume that f 3 = x . The line
3
3
Z(x 3 ) is parameterized by θ where θ(s, t)=(s, t, 0). Assume that the polynomial
f 4 (θ) has r distinct roots with multiplicities m 1 ,...,m r . (As before f 4 (θ) ≡ 0 if and