Page 46 - Geometric Modeling and Algebraic Geometry

2 Some Covariants Related to Steiner Surfaces 43
Dual conic to the preimage of Π.

3
In [13], parag. 377 is shown a covariant Ψ(q 1 ,q 2 ,λ) of forms on C (q 1 and q 2
quadratic, λ linear), whose vanishing is a necessary and sufficient condition for the
traces of the conics of equations q 1 (x)=0 and q 2 (x)=0 on the line of equation
λ(x)=0 to be a harmonic system of points.
Set

Φ 14 = ∂Φ 5 Ψ(f i ,f j ,λ) (2.31)
dy i dy j
i,j
where
λ(x)= λ 0 x 0 + λ 1 x 1 + λ 2 x 2 .
Then Φ 14 is a covariant of degree 8 and type Pol ((C ) ). One checks that Φ 14 (f)=
3 ∗
2
0 is an equation for the conic of (CP ) dual to the conic of equation Φ 13 (f)=0 of
2 ∗
CP . Note that the equation we find this way has lower degree than the one obtained
2
by computing the comatrix of the matrix of Φ 13 (f) (that would have degree 32).
Quadrilateral, preimage of the four torsal conics.
The union of the four torsal conics is also the intersection between the Associated
Quadric (defined by Φ 5 (f)=0) and the Steiner surface. Thus, its preimage is also
the preimage of the quadric.
Substitute y i with f i in Φ 5 , this gives a new covariant Φ 15 of degree 8 and type
Pol (C ):
3
4
Φ 15 (f)(x)= Φ 5 (f)(f(x)). (2.32)
The zero locus of Φ 15 (f) in CP is the quadrilateral, preimage of the union of the
2
torsal conics.

2.6 Application: Equations and inequalities defining the types
of Steiner surfaces

We want to recognize the orbits in U, that is the orbits of parameterizations of quartic
surfaces (from those of surfaces of smaller degree), and next to discriminate between
these orbits.
We consider the first task. After [2] (Proposition 2 and Proposition 5), there are
three cases to rule out. The first case is when the parameterization [f] admits a base
point (i.e. the f i ’s have a common zero in CP ). The second case corresponds to the
2
orbit of the parameterization
(x : x : x : x 1 x 2 ). (2.33)
2
2
2
0 1 2
The Zariski closure of its image is a quadric. The third case is the case when the
Zariski closure of the image of the parametrization is a plane. A necessary and suffi-
cient condition for being in the first case is the identical vanishing of Φ 4 (f), which