Page 40 - Geometric Modeling and Algebraic Geometry

2 Some Covariants Related to Steiner Surfaces 37
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to its type (its space of values ). Note that the definition of this geometric object
will be valid only in the case when [f] parameterizes a Steiner surface.
We will meet covariants of the following types:

3
• Type Pol (C ): such a covariant C associates to [f] a surface in CP (the zero
4
n
locus of C(f)).
• Type Pol (C ): such a covariant associates to [f] a curve in CP .
2
3
n
4 ∗
3 ∗
• Type Pol ((C ) ): such a covariant associates to [f] a surface in (CP ) . If this
n
3 ∗
surface is decomposable, that is a union of hyperplanes of (CP ) , then it also
represents a finite collection of points in CP (the points corresponding to the
3
hyperplanes by duality).
3 ∗
2 ∗
• Type Pol ((C ) ): such a covariant associates to [f] a curve in (CP ) .Ifthis
n
curve is decomposable, then it also represents a finite collection of points in CP .
2
• Type some space of functions Pol (W, V ) between spaces W, V among C , C 4
3
n
and their duals. Then the covariant associates to [f] some family of curves or
surfaces in P(V ) parameterized by P(W).
∗
• Type C: such a homogeneous covariant is just an invariant for the group
SL(3, C) × SL(4, C). We will see that there is essentially only one invariant.
The geometric objects attached to some of the covariants we will present will be
clear from their construction; for the rest, they can be found merely by evaluating the
covariant on the representative of Orbit I C :

2 x 1 x 2 :2 x 0 x 2 :2 x 0 x 1 : x 0 + x 1 + x 2 . (2.11)
2
2
2
Table 2.2 recapitulates the list of covariants that will be now presented individu-
ally. The reader will find Maple procedures implementing the formulas that follow
on the web page:
http://emmanuel.jean.briand.free.fr/publications/steiner/
2.5.2 Derivation of the covariants
Here we suppose that [f] is in I C , that is its image S(f) in CP is a complex Steiner
3
surface.
For each covariant we indicate its type, and its degree with respect to the coeffi-
cients of the f i ’s.
Tangent plane at the image of a point.
Given a generic point [x] in the parameter space CP , we can consider the tangent
2
plane to the Steiner surface S(f) at its image by [f]. It has equation Φ 1 (f)(x)=0,
where
Strictly speaking, the type should mention also the action of G on this space. In all the
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cases we will meet, this action is a canonical action of G on the space, or its product by
some powers of the determinants of θ ∈ GL(3, R) and ρ ∈ GL(4, R). These powers are
easily determined from the degree of the covariant.