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2 Some Covariants Related to Steiner Surfaces 33
2.6, presents the application of these covariants to the discrimination of classes of
parameterizations.

2.2 Orbits of quadratic parameterizations of quartics

A quadratic rational map from RP to RP is determined by a homogeneous
3
2
quadratic map f from R to R , that can be presented as a family of four real ternary
4
3
quadratic forms:
f =(f 0 (x 0 ,x 1 ,x 2 ),f 1 (x 0 ,x 1 ,x 2 ),f 2 (x 0 ,x 1 ,x 2 ),f 3 (x 0 ,x 1 ,x 2 )) . (2.3)
Denote with F the space of all the quadruples of real ternary quadratic forms. Then,
more precisely, quadratic rational maps from RP to RP can be identified with
2
3
the elements of F considered modulo scalar multiplication, i.e. the projective space
P(F).For f ∈F, we will denote with [f] the corresponding element of P(F).
Now the group GL(3, R) acts naturally on R (and RP ), and thus on F (and
2
3
P(F)). The action on F is as follows: for θ ∈ GL(3, R),
θ(f)= f ◦ θ −1 . (2.4)
The induced action on P(F) corresponds to linear reparameterizations. There is also
a natural action of the group GL(4, R) on R (and RP ), and thus on F (and P(F)):
3
4
for ρ ∈ GL(4, R),
ρ(f)= ρ ◦ f. (2.5)
We have thus an action of GL(3, R) × GL(4, R) on F (and P(F)). In the sequel, we
will denote this group with G.
In P(F), the subset U of those projective parameterizations with the property that
4
the Zariski closure of their image is a surface of degree 4 exactly, is invariant under
G. It is also a Zariski dense open set. As said in the introduction, the decomposition
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of U into orbits is known ; see [2, 7] and [8]. There are only six orbits. Table 2.1
provides the list of the orbits, with a representative for each.
Let us say a word about the connection between this problem and the analogous
problem in the complex setting. Denote with F C the complexification of F: that
is the space of families of four complex quadratic forms. Then P(F C ) represents
the space of quadratic rational maps from the complex projective plane, CP to the
2
complex projective three–dimensional space, CP . Let U C be the subset of those
3
parameterizations whose image is a quartic surface. Then U is the trace of U C on
P(F). This means that U = U C ∩ P(F).
Let G C = GL(3, C)×GL(4, C). This group acts naturally on F C and P(F C ), and
also on U C . The classification of the orbits of P(F) under G is obtained by refining
the classification of P(F C ) into orbits under G C (see [1] for a modern reference about

We consider the set–theoretical image, and rule out the cases when the Zariski closure of
4
the image is a double quadric (case 7 in Proposition 5 of [2]) or a plane counted four times.
The determination of the orbits outside U is a different problem. See the references in [7].
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