Page 77 - Geometric Modeling and Algebraic Geometry

74 P. H. Johansen et al.
only if the monoid is reducible.) Then the type of the singularity at O will be of the

V series [3, p. 267]. The integers m 1 ,...,m r are constant under right equivalence
over . Note that one can construct examples of monoids that are right equivalent
over , but not over (see Figure 4.4).

Fig. 4.4. The monoids Z(z + xy + x y) and Z(z + xy − x y) are right equivalent over
3
3
3
3
3
3
but not over .
The tangent cone is singular everywhere, so there can be no other singularities
on the monoid.
Case 9. The tangent cone is a smooth cubic curve, and we write f 3 = x + x +
3
3
1 2
x +3ax 1 x 2 x 3 where a = −1. This is a one-parameter family of elliptic curves,
3
3
3
so we cannot use the parameterization technique of the other cases. The singularity
at O will be a P 8 singularity (cf. [3, p. 185]), and other singularities correspond to
intersections between Z(f 3 ) and Z(f 4 ), as described by Proposition 6.
To classify the possible configurations of singularities on a monoid with a non-
singular (projective) tangent cone, we need to answer the following question: For
r
any positive integers m 1 ,...,m r such that m i =12, does there, for some
i=1
a ∈ \{−1}, exist a polynomial f 4 with real coefficients such that Z(f 3 ,f 4 )=
(f 3 ,f 4 )= m i for i =1,...,r? Rohn [15, p. 63] says
2
{p 1 ,...,p r }∈ P ( ) and I p i
that one can always find curves Z(f 3 ), Z(f 4 ) with this property. Here we shall show
that for any a ∈ \{−1} we can find a suitable f 4 .
In fact, in almost all cases f 4 can be constructed as a product of linear and
quadratic terms in a simple way. The difficult cases are (m 1 ,m 2 )=(11, 1),
(m 1 ,m 2 ,m 3 )=(8, 3, 1), and (m 1 ,m 2 )=(5, 7). For example, the case where
2
(m 1 ,m 2 ,m 3 )=(3, 4, 5) can be constructed as follows: Let f 4 =
1
2
where
1
3
and
2 define tangent lines at inflection points p 1 and p 3 of Z(f 3 ).Let
3 define a line
that intersects Z(f 3 ) once at p 3 and twice at another point p 2 . Note that the points
p 1 , p 2 and p 3 can be found for any a ∈ \{−1}.