Page 102 - Geometric Modeling and Algebraic Geometry

100 T.-H. Lˆ e and A. Galligo
∗
∗
This condition defines a scroll F(2, 2) in (P ) that we call the dual scroll of F(2, 2).
5
5 ∗
The parametric equations of the scroll F(2, 2) in the dual space (P ) were given
∗
above.
6.3.4 Intersection of Π A and F(2, 2) ∗
∗
By replacing the parametric equations of F(2, 2) in the implicit equation of Π A ,we
see that Π A ∩ F(2, 2) is given by the intersection of two curves of bidegree (1,2) in
∗
the parameter space P × P :
1
1

ϕ 1 (t, u)= A 1 − 2B 1 u + C 1 u − D 1 t +2E 1 tu − F 1 u t =0
2
2
ϕ 2 (t, u)=2B 2 u + C 2 u − D 2 t +2E 2 tu − F 2 u t =0.
2
2
We have two cases: either ϕ 1 (t, u) ∩ ϕ 2 (t, u) is finite (4 points) or infinite. We first
consider the generic cases, i.e. the intersection contains 4 distinct points (t k ,u k ),
k =1,..., 4 and t k = t j , u k = u j if k = j. This will give a classification of the
maps of bidegree (1,2) up to change of coordinates and a set of normal forms.
6.4 The generic case

We first recall the result for the generic complex case (see more details in the article
[13]).

6.4.1 The generic complex case

Generically, ϕ 1 (t, u) ∩ ϕ 2 (t, u) contains 4 distinct points; they correspond to 4 tan-
gent planes. They are tangent to all conics of S, along a special torsal line.

We can choose these 4 tangent planes in P to be the planes of coordinates (X =
3
0), (Y =0), (Z =0), (T =0).

We proved in [13] that, after a suitable change of coordinates and change of pa-
rameters, the surface S admits the parametric representation:
⎧
X = tu 2
⎪
⎪
Y =(t − s)(u − v) 2
⎨
⎪ Z =(t − as)(u − bv) 2
⎪
T = sv 2
⎩
This normal form depends on two moduli a and b. Moreover, the singular locus of
the surface is a twisted cubic and has parametric equations (c.f [13]): (t : s) −→
(X : Y : Z : T)
⎧
⎪ X = abt(bt − t − bs + as) 2
⎪
Y =(a − 1)(as − bt)(bt − b t + b t + bats − bts − ats + as − bas )
⎨
2
2
2 2
2
2
⎪ Z = a(a − 1)b(bs − t)(bt − t + bts + ats − bats − b ts − bas + b as )
2
2
2
2
2
2
⎪
T =(at − bt − as + bas) s
⎩
2