Page 74 - Geometric Modeling and Algebraic Geometry

4 Monoid Hypersurfaces 71
monoid is reducible if and only if f 4 (θ 1 ) ≡ 0 or f 4 (θ 2 ) ≡ 0. Consider two nonzero
polynomials
q 1 = b 0 s + b 1 s t + b 2 s t + b 3 st + b 4 t 4
2 2
4
3
3
q 2 = c 0 s + c 1 s t + ··· + c 7 st + c 8 t .
7
8
7
8
Now (λ 1 q 1 ,λ 2 q 2 )=(f 4 (θ 1 ),f 4 (θ 2 )) for some f 4 if and only if λ 1 b 0 = λ 2 c 0 and
λ 1 b 1 = λ 2 c 1 . As before, only the cases where λ 1 ,λ 2 =0 are interesting. We see
that (λ 1 q 1 ,λ 2 q 2 )=(f 4 (θ 1 ),f 4 (θ 2 )) for some λ 1 ,λ 2 =0 if and only if the following
hold:
• b 0 =0 ↔ c 0 =0 and b 1 =0 ↔ c 1 =0
• b 0 c 1 = b 1 c 0 .
The classification of other singularities (than O) is very similar to the previous
case. Roots of f 4 (θ 1 ) and f 4 (θ 2 ) away from (1 : 0) correspond to intersections of
Z(f 3 ) and Z(f 4 ) away from the singular point of Z(f 3 ), and when one such inter-
section is multiple, there is a corresponding singularity on the monoid.
Now assume (λ 1 q 1 ,λ 2 q 2 )=(f 4 (θ 1 ),f 4 (θ 2 )) for some λ 1 ,λ 2 =0 and some
f 4 .If b 0 =0 (equivalent to c 0 =0) then j 0 = k 0 =0 and λ 1 /λ 2 = c 0 /b 0 .
If b 0 = c 0 =0 and b 1 =0 (equivalent to c 1 =0), then j 0 = k 0 =1, and
λ 1 /λ 2 = c 1 /b 1 .If b 0 = b 1 = c 0 = c 1 =0, then j 0 ,k 0 > 1 and any value of
λ 1 /λ 2 will give (λ 1 q 1 ,λ 2 q 2 ) of the form (f 4 (θ 1 ),f 4 (θ 2 )) for some f 4 . Thus we get
a one-dimensional family of monoids for this choice of q 1 and q 2 .
Now consider the possible configurations of other singularities on the monoid.

Assume that j ≤ 8 and k ≤ 4 are nonnegative integers such that j 0 > 0 ↔
0 0
k 0 > 0 and j 0 > 1 ↔ k 0 > 1. For any set of multiplicities m 1 ,...,m r with

m 1 + ··· + m r =4 − k and m ,...,m with m + ··· + m =8 − j , there
r
r
0 1 1 0
exists a polynomial f 4 with real coefficients such that f 4 (θ 1 ) has real roots away
from (1 : 0) with multiplicities m 1 ,...,m r , and f 4 (θ 2 ) has real roots away from
(1 : 0) with multiplicities m ,...,m . Furthermore, for this f 4 we have k 0 = k

r
1 0
and j 0 = j . Proposition 6 will give the singularities that occur in addition to O.

0
This completes the classification of the singularities on a quartic monoid (other
than O) when the tangent cone is a conic plus a tangent.
Case 5. The tangent cone is three general lines, and we assume f 3 = x 1 x 2 x 3 .
For each f 4 we associate six integers,
k 2 := I (1:0:0) (f 4 ,x 2 ),l 1 := I (0:1:0) (f 4 ,x 1 ),m 1 := I (0:0:1) (f 4 ,x 1 ),
k 3 := I (1:0:0) (f 4 ,x 3 ),l 3 := I (0:1:0) (f 4 ,x 3 ),m 2 := I (0:0:1) (f 4 ,x 2 ).
Now k 2 > 0 ⇔ k 3 > 0, l 1 > 0 ⇔ l 3 > 0, and m 1 > 0 ⇔ m 2 > 0. If both k 2 and
k 3 are greater than 1, then the monoid has a singular line, a case we have excluded.
The same goes for the pairs (l 1 ,l 3 ) and (m 1 ,m 2 ).
When the monoid does not have a singular line, we define j k = max(k 2 ,k 3 ),
j l = max(l 1 ,l 3 ) and j m =max(m 1 ,m 2 ).If j k ≤ j l ≤ j m , then [19] gives that O
singularity.
is a T 4+j k ,4+j l ,4+j m