Page 78 - Geometric Modeling and Algebraic Geometry

4 Monoid Hypersurfaces 75
The case (m 1 ,m 2 )=(11, 1) is also possible for every a ∈ \{−1}. For any
point p on Z(f 3 ) there exists an f 4 such that I p (f 3 ,f 4 ) ≥ 11. For all except a finite
number of points, we have equality [11], so the case (m 1 ,m 2 )=(11, 1) is possible
for any a ∈ \{−1}. The case (m 1 ,m 2 ,m 3 )=(8, 3, 1) is similar, but we need to
let f 4 be a product of the tangent at an inflection point with another cubic.
The case (m 1 ,m 2 )=(5, 7) is harder. Let a =0. Then we can construct a conic
C that intersects Z(f 3 ) with multiplicity five in one point and multiplicity one in an
inflection point, and choosing Z(f 4 ) as the union of C and twice the tangent line
through the inflection point will give the desired example. The same can be done
for a = −4/3. By using the computer algebra system SINGULAR [6] we can show
that these constructions can be continuously extended to any a ∈ \{−1}. This
completes the classification of singularities on a monoid when the tangent cone is
smooth.
In the Cases 3, 5, and 6, not all real equations of a given type can be transformed
to the chosen forms by a real transformation.
In Case 3 the conic may not intersect the line in two real points, but rather in two
complex conjugate points. Then we can assume f 3 = x 3 (x 1 x 3 + x + x ), and the
2
2
1 2
singular points are (1 : ±i :0). For any real f 4 , we must have
I (1:i:0) (x 1 x 3 + x + x ,f 4 )=I (1:−i:0) (x 1 x 3 + x + x ,f 4 )
2
2
2
2
1 2 1 2
and
I (1:i:0) (x 3 ,f 4 )=I (1:−i:0) (x 3 ,f 4 ),
so only the cases where j 0 = j 1 and k 0 = k 1 are possible. Apart from that, no other
restrictions apply.
In Case 5, two of the lines can be complex conjugate, and we assume f 3 =
x 3 (x + x ). A configuration from the previous analysis is possible for real coeffi-
2
2
1 2
cients of f 4 if and only if m 1 = m 2 , k 2 = l 1 , and k 3 = l 3 . Furthermore, only the
singularities that correspond to the line Z(x 3 ) will be real.
In Case 6, two of the lines can be complex conjugate, and then we may assume
f 3 = x + x .Now,if j 3 denotes the intersection number of Z(f 4 ) with the real line
3
3
2 3
Z(x 2 + x 3 ), precisely the cases where j 1 = j 2 are possible, and only intersections
with the line Z(x 2 + x 3 ) may contribute to real singularities.
This concludes the classification of real and complex singularities on real
monoids of degree 4.
Remark. In order to describe the various monoid singularities, Rohn [15] com-
putes the “class reduction” due to the presence of the singularity, in (almost) all
cases. (The class is the degree of the dual surface [14, p. 262].) The class reduction
is equal to the local intersection multiplicity of the surface with two general polar
surfaces. This intersection multiplicity is equal to the sum of the Milnor number and
the Milnor number of a general plane section through the singular point [20, Cor. 1.5,
p. 320]. It is not hard to see that a general plane section has either a D 4 (Cases 1–6,
9), D 5 (Case 7), or E 6 (Case 8) singularity. Therefore one can retrieve the Milnor
number of each monoid singularity from Rohn’s work.