Page 67 - Geometric Modeling and Algebraic Geometry

64 P. H. Johansen et al.

d
Fig. 4.2. The surface defined by F = x 0(x 1x d−2 + x d−1 )+ x 1 for d =4.
2 3

4.4 Quartic monoid surfaces

Every cubic surface with isolated singularities is a monoid. Both smooth and singular
cubic surfaces have been studied extensively, most notably in [16], where real cubic
surfaces and their singularities were classified, and more recently in [18], [4], and
[8]. The site [9] contains additional pictures and references.
In this section we shall consider the case d =4. The classification of real and
complex quartic monoid surfaces was started by Rohn [15]. (In addition to consid-
ering the singularities, Rohn studied the existence of lines not passing through the
triple point, and that of other special curves on the monoid.) In [19], Takahashi,
Watanabe, and Higuchi described the singularities of such complex surfaces. The
monoid singularity of a quartic monoid is minimally elliptic [21], and minimally el-
liptic singularities have the same complex topological type if and only if their dual
graphs are isomorphic [10]. In [10] all possible dual graphs for minimally elliptic
singularities are listed, along with example equations.
Using Arnold’s notation for the singularities, we use and extend the approach of
Takahashi, Watanabe, and Higuchi in [19].
Consider a quartic monoid surface, X = Z(F), with F = x 0 f 3 +f 4 . The tangent
cone, Z(f 3 ), can be of one of nine (complex) types, each needing a separate analysis.
For each type we fix f 3 , but any other tangent cone of the same type will be pro-
jectively equivalent (over the complex numbers) to this fixed f 3 . The nine different
types are:

1. Nodal irreducible curve, f 3 = x 1 x 2 x 3 + x + x .
3
3
2 3
2. Cuspidal curve, f 3 = x − x x 3 .
2
3
1 2
3. Conic and a chord, f 3 = x 3 (x 1 x 2 + x )
2
3
4. Conic and a tangent line, f 3 = x 3 (x 1 x 3 + x ).
2
2
5. Three general lines, f 3 = x 1 x 2 x 3 .
6. Three lines meeting in a point, f 3 = x − x 2 x 2
3
2 3