Page 59 - Geometric Modeling and Algebraic Geometry

56 P. H. Johansen et al.
gave a fairly complete description of all possible cases. He also remarked [15, p. 56]
that some of his results on quartic monoids hold for monoids of arbitrary degree; in
particular, we believe he was aware of many of the results in Section 4.3. Takahashi,
Watanabe, and Higuchi [19] classify complex quartic monoid surfaces, but do not
refer to Rohn. (They cite Jessop [7]; Jessop, however, only treats quartic surfaces
with double points and refers to Rohn for the monoid case.) Here we aim at giving
a short description of the possible singularities that can occur on quartic monoids,
with special emphasis on the real case.

4.2 Basic properties

¯ n n
Let k be a field, let k denote its algebraic closure and P := P ¯ k the projective n-
¯
n
space over k. Furthermore we define the set of k-rational points P (k) as the set of
points that admit representatives (a 0 : ··· : a n ) with each a i ∈ k.
¯
For any homogeneous polynomial F ∈ k[x 0 ,...,x n ] of degree d and point p =
n
(p 0 : p 1 : ··· : p n ) ∈ P we can define the multiplicity of Z(F) at p. We know that
p r =0 for some r, so we can assume p 0 =1 and write
d
d−i
F = x f i (x 1 − p 1 x 0 ,x 2 − p 2 x 0 ,...,x n − p n x 0 )
0
i=0
where f i is homogeneous of degree i. Then the multiplicity of Z(F) at p is defined
to be the smallest i such that f i =0.
¯
Let F ∈ k[x 0 ,...,x n ] be of degree d ≥ 3. We say that the hypersurface X =
n
Z(F) ⊂ P is a monoid hypersurface if X is irreducible and has a singular point of
multiplicity d − 1.
In this article we shall only consider monoids X = Z(F) where the singular
n
point is k-rational. Modulo a projective transformation of P over k we may – and
shall – therefore assume that the singular point is the point O =(1:0: ··· :0).
Hence, we shall from now on assume that X = Z(F), and
F = x 0 f d−1 + f d ,

where f i ∈ k[x 1 ,...,x n ] ⊂ k[x 0 ,...,x n ] is homogeneous of degree i and f d−1 =
0. Since F is irreducible, f d is not identically 0, and f d−1 and f d have no common
(non-constant) factors.
The natural rational parameterization of the monoid X = Z(F) is the map
θ F : P n−1 → P n

given by
θ F (a)=(f d (a): −f d−1 (a)a 1 : ... : −f d−1 (a)a n ),
for a =(a 1 : ··· : a n ) such that f d−1 (a) =0 or f d (a) =0.