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2 Some Covariants Related to Steiner Surfaces 35
For G = G acting on W = F, a polynomial covariant for the action of G on F
is a polynomial map from F to some G–module such that
C(ρ ◦ f ◦ θ −1 )=(ρ, θ)(C(f)) (2.8)

for all θ ∈ GL(3, R) and all ρ ∈ GL(4, R).
Note that the zero set of any covariant is a G–invariant set, that is a union of
orbits.
We finish this section with some remarks. The covariants for F under G are
essentially the same as those of F C under G C : the former are obtained by complex-
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ification of the latter . From a classical theorem of Invariant Theory (see [12]), we
know that the homogeneous covariants separate the orbits of P(F C ) under G C :this
means that for any two orbits O 1 and O 2 , there exists some homogeneous covariant
vanishing on O 1 and not on O 2 ,or vice–versa. On the contrary, there is no guaran-
tee in advance that we can separate the orbits of P(F) under G using equations and
inequalities involving only the covariants. We will be able to do it in Section 2.6 by
using some derived objects.

2.4 Some elements of the geometry of the Steiner surface
To each of the covariants we will introduce is attached a simple geometric object
associated to the quadratic parameterizations of the complex Steiner surface. This is,
actually, what will guide us in the construction of the covariants.
We now introduce the main features of the Steiner surface (they can be found
in [14], parag. 554a). For f ∈F, denote with S(f) the associated complex Steiner
surface, that is the image of CP under [f]. Then:
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• It is a quartic (its implicit equation has degree 4).
• Its singular locus is the union of three lines, that are double lines. They are con-
current: their intersection is the unique triple point of the Steiner surface.
• The intersection of S(f) with a tangent plane is a quartic curve that either de-
composes as the union of two conics intersecting at four points, or as a double
conic. The latter situation happens only for four tangent planes, that Salmon calls
tropes. In the former situation, one of the four intersection points is the point of
tangency; the three remaining points are the intersections of the plane with each
of three double lines.
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• Each trope is tangent to the Steiner surface along a conic, called a torsal conic .
There are thus four torsal conics.
• There is a unique quadric going through the four torsal conics. Let us call it the
Associated Quadric.
3 ∗
• The dual (or “reciprocal”) surface to S(f) (the surface of (CP ) that is the
Zariski closure of the set of all tangent planes to S(f)) is a cubic surface, known
as the Cayley Cubic Surface (see [14]).
For such issues of field of definition, see [11].
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This is called a parabolic conic in [7].
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