Page 61 - Geometric Modeling and Algebraic Geometry

58 P. H. Johansen et al.
4.3 Monoid surfaces

In the case of a monoid surface, the parameterization has a finite number of base
points. From Lemma 1 (ii) we know that all singularities of the monoid other than
O, are on lines L a corresponding to these points. In what follows we will develop the
theory for singularities on monoid surfaces — most of these results were probably
known to Rohn [15, p. 56].
We start by giving a precise definition of what we shall mean by a monoid sur-
face.

Definition 2. For an integer d ≥ 3 and a field k of characteristic 0 the polynomials
f d−1 ∈ k[x 1 ,x 2 ,x 3 ] d−1 and f d ∈ k[x 1 ,x 2 ,x 3 ] d define a normalized non-degenerate
monoid surface Z(F) ⊂ P , where F = x 0 f d−1 + f d ∈ k[x 0 ,x 1 ,x 2 ,x 3 ] if the
3
following hold:
(i) f d−1 ,f d =0
(ii) gcd(f d−1 ,f d )=1
(iii) The curves Z(f d−1 ) ⊂ P and Z(f d ) ⊂ P have no common singular point.
2
2
The curves Z(f d−1 ) ⊂ P and Z(f d ) ⊂ P are called respectively the tangent cone
2
2
and the intersection with infinity.
Unless otherwise stated, a surface that satisfies the conditions of Definition 2
shall be referred to simply as a monoid surface.
Since we have finitely many base points b and each line L b contains at most
one singular point in addition to O, monoid surfaces will have only finitely many
singularities, so all singularities will be isolated. (Note that Rohn included surfaces
with nonisolated singularities in his study [15].) We will show that the singularities
other than O can be classified by local intersection numbers.
Definition 3. Let f, g ∈ k[x 1 ,x 2 ,x 3 ] be nonzero and homogeneous. Assume p =
(p 1 : p 2 : p 3 ) ∈ Z(f, g) ⊂ P , and define the local intersection number
2
¯
k[x 1 ,x 2 ,x 3 ] m p
I p (f, g)=lg ,
(f, g)
¯
where k is the algebraic closure of k, m p =(p 2 x 1 −p 1 x 2 ,p 3 x 1 −p 1 x 3 ,p 3 x 2 −p 2 x 3 )
is the homogeneous ideal of p, and lg denotes the length of the local ring as a module
over itself.

Note that I p (f, g) ≥ 1 if and only if f(p)= g(p)=0. When I p (f, g)=1 we say
that f and g intersect transversally at p. The terminology is justified by the following
lemma:
Lemma 4. Let f, g ∈ k[x 1 ,x 2 ,x 3 ] be nonzero and homogeneous and p ∈ Z(f, g).
Then the following are equivalent:
(i) I p (f, g) > 1