Page 60 - Geometric Modeling and Algebraic Geometry

4 Monoid Hypersurfaces 57
The set of lines through O form a P n−1 . For every a =(a 1 : ··· : a n ) ∈ P n−1 ,
the line
L a := {(s : ta 1 : ... : ta n )|(s : t) ∈ P } (4.1)
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intersects X = Z(F) with multiplicity at least d − 1 in O.If f d−1 (a) =0 or
f d (a) =0, then the line L a also intersects X in the point
θ F (a)=(f d (a): −f d−1 (a)a 1 : ... : −f d−1 (a)a n ).

Hence the natural parameterization is the “inverse” of the projection of X from the
point O. Note that θ F maps Z(f d−1 ) \ Z(f d ) to O. The points where the parameter-
ization map is not defined are called base points, and these points are precisely the
common zeros of f d−1 and f d . Each such point b corresponds to the line L b con-
tained in the monoid hypersurface. Additionally, every line of type L b contained in
the monoid hypersurface corresponds to a base point.
Note that Z(f d−1 ) ⊂ P n−1 is the projective tangent cone to X at O, and that
Z(f d ) is the intersection of X with the hyperplane “at infinity” Z(x 0 ).
Assume P ∈ X is another singular point on the monoid X. Then the line L
through P and O has intersection multiplicity at least d − 1+2 = d +1 with X.
Hence, according to Bezout’s theorem, L must be contained in X, so that this is only
possible if dim X ≥ 2.
By taking the partial derivatives of F we can characterize the singular points of
X in terms of f d and f d−1 :
Lemma 1. Let ∇ =( ∂ ,..., ∂ ) be the gradient operator.
∂x n
∂x 1
n
(i) A point P =(p 0 : p 1 : ··· : p n ) ∈ P is singular on Z(F) if and only if
f d−1 (p 1 ,...,p n )=0 and p 0 ∇f d−1 (p 1 ,...,p n )+ ∇f d (p 1 ,...,p n )=0.
(ii) All singular points of Z(F) are on lines L a where a is a base point.
(iii) Both Z(f d−1 ) and Z(f d ) are singular in a point a ∈ P n−1 if and only if all
points on L a are singular on X.
(iv) If not all points on L a are singular, then at most one point other than O on L a
is singular.
Proof. (i) follows directly from taking the derivatives of F = x 0 f d−1 + f d , and (ii)
follows from (i) and the fact that F(P)=0 for any singular point P. Furthermore, a
point (s : ta 1 : ... : ta n ) on L a is, by (i), singular if and only if
s∇f d−1 (ta)+ ∇f d (ta)= t d−1 (s∇f d−1 (a)+ t∇f d (a)) = 0.

This holds for all (s : t) ∈ P if and only if ∇f d−1 (a)= ∇f d (a)=0. This proves
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(iii). If either ∇f d−1 (a) or ∇f d−1 (a) are nonzero, the equation above has at most
one solution (s 0 : t 0 ) ∈ P in addition to t =0, and (iv) follows.
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Note that it is possible to construct monoids where F ∈ k[x 0 ,...,x n ], but where
no points of multiplicity d − 1 are k-rational. In that case there must be (at least) two
such points, and the line connecting these will be of multiplicity d − 2. Furthermore,
the natural parameterization will typically not induce a parameterization of the k-
rational points from P n−1 (k).