Page 66 - Geometric Modeling and Algebraic Geometry

4 Monoid Hypersurfaces 63
For a real, monoid Corollary 7 implies that we can have at most d(d − 1) real
1
2
singularities in addition to O. We can show that the bound is sharp by a simple
construction:
Example. To construct a monoid with the maximal number of real singularities, it
is sufficient to construct two affine real curves in the xy-plane defined by equations
f d−1 and f d of degrees d−1 and d such that the curves intersect in d(d−1)/2 points
with multiplicity 2. Let m ∈{d − 1,d} be odd and set
m
2iπ 2iπ
f m = ε − x sin + y cos +1 .
m m
i=1
For ε> 0 sufficiently small there exist at least m+1 radii r> 0, one for each root
2
of the univariate polynomial f m | x=0 , such that the circle x + y − r intersects
2
2
2
f m in m points with multiplicity 2.Let f 2d−1−m be a product of such circles. Now
the homogenizations of f d−1 and f d define a monoid surface with 1+ d(d − 1)
1
2
singularities. See Figure 4.1.

Fig. 4.1. The curves f m for m =3, 5 and corresponding circles

Proposition 6 and Bezout’s theorem imply that the maximal Milnor number of
a singularity other than O is d(d − 1) − 1. The following example shows that this
bound can be achieved on a real monoid:
d
Example. The surface X ⊂ P defined by F = x 0 (x 1 x d−2 +x d−1 )+x has exactly
3
1
2 3
two singular points. The point (1 :0:0:0) is a singularity of multiplicity d − 1
with Milnor number µ =(d − 3d +1)(d − 2), while the point (0 :0:1:0) is an
2
A d(d−1)−1 singularity. A picture of this surface for d =4 is shown in Figure 4.2.