Page 71 - Geometric Modeling and Algebraic Geometry

68 P. H. Johansen et al.
when m ≥ 2 any choice of p 1 ,...,p r and m 1 ,...,m r with m 1 + ··· + m r =
12 − m corresponds to a four dimensional space of equations f 4 that gives this set

of roots and multiplicities in f 4 (θ).If f is one such f 4 , then any other is of the form
4
λf +
f 3 for some constant λ =0 and linear form
. All of these give monoids that

4
are projectively equivalent.
When m =0, we write p i =(α i : β i ) for i =1,...,r. The condition b 0 = b 12
on the coefficients of q translates to
α m 1 ··· α m r = β m 1 ··· β m r . (4.3)
r
r
1 1
This means that any choice of parameter points (α 1 : β 1 ),..., (α r : β r ) and mul-
tiplicities m 1 ,...,m r with m 1 + ··· + m r =12 that satisfy condition (4.3) cor-
responds to a four dimensional family λf +
f 3 , giving a unique monoid up to

4
projective equivalence.
For example, we can have an A 11 singularity only if f 4 (θ) is of the form
(αs − βt) . Condition (4.3) implies that this can only happen for 12 parameter
12
points, all of the form (1 : ω), where ω 12 =1. Each such parameter point (1 : ω)
corresponds to a monoid uniquely determined up to projective equivalence. How-
ever, since there are six projective transformations of the plane that maps Z(f 3 ) onto
itself, this correspondence is not one to one. If ω 12 = ω 12 =1, then ω 1 and ω 2 will
1 2
correspond to projectively equivalent monoids if and only if ω = ω or ω ω =1.
3
3
3
3
1 2 1 2
This means that there are three different quartic monoids with one T 3,3,4 singularity
and one A 11 singularity. One corresponds to those ω where ω =1, one to those ω
3
where ω = −1, and one to those ω where ω = −1. The first two of these have real
6
3
representatives, ω = ±1.
It easy to see that for any set of multiplicities m 1 + ··· + m r =12, we can find
real points p 1 ,...,p r such that condition (4.3) is satisfied. This completely classifies
the possible configurations of singularities when f 3 is an irreducible nodal curve.
Case 2. The tangent cone is a cuspidal curve, and we can assume f 3 (x 1 ,x 2 ,x 3 )=
x − x x 3 . The cuspidal curve is singular at (0 :0:1) and can be parameterized by
2
3
1 2
θ as a map from P to P where θ(s, t)=(s t, s ,t ). The intersection numbers are
3
2
2
1
3
determined by the degree 12 polynomial f 4 (θ). As in the previous case, f 4 (θ) ≡ 0
if and only if f 3 is a factor of f 4 , and we will assume this is not the case. The mul-
tiplicity m of the factor s in f 4 (θ) determines the type of singularity at O.If m =0
(no factor s), then O is a Q 10 singularity. If m =2 or m =3, then O is of type
Q 9+m .If m> 3, then (0 :0:1) is a singular point on Z(f 4 ), so the monoid has a
singular line and is not considered in this article. Also, m =1 is not possible, since
f 4 (θ(s, t)) = f 4 (s t, s ,t ) cannot contain st 11 as a factor.
2
3
3
For each m =0, 2, 3 we can analyze the possible configurations of other sin-
gularities on the monoid. Similarly to the previous case, any choice of parameter
points p 1 ,...,p r ∈ P \{(0 : 1)} and positive multiplicities m 1 ,...,m r with
1

m i =12 − m corresponds, up to a nonzero constant, to a unique degree 12 poly-
nomial q.
When m =2 or m =3, for any choice of parameter values and associated mul-

tiplicities, we can find a four dimensional family f 4 = λf +
f 3 with the prescribed
4
roots in f 4 (θ). As before, the family gives projectively equivalent monoids.