Page 44 - Geometric Modeling and Algebraic Geometry

2 Some Covariants Related to Steiner Surfaces 41
Φ 7 =Disc (α 0 f 0 + α 1 f 1 + α 2 f 2 + α 3 f 3 ) . (2.21)

4 ∗
The object obtained this way, Φ 7 , is a covariant. It has degree 3 and type Pol ((C ) ).
3
The zero locus of Φ 7 (f) is the dual surface to S(f).
Triple point.

4 ∼
A covariant of degree 9 and type C = Pol ((C ) ) is produced by contraction of
4 ∗
1
Φ 7 and Φ 5 :
2 2
Φ 8 = ∂ Φ 5 ∂ Φ 7 . (2.22)
dy i dy j dα i dα j
i,j
Write
Φ 8 (f)= τ 0 α 0 + τ 1 α 1 + τ 2 α 2 + τ 3 α 3 . (2.23)
Then the associated geometric object is a point (τ 0 : τ 1 : τ 2 : τ 3 ) of CP . One checks
3
that this is exactly the triple point of S(f).

Discriminant.
By evaluating Φ 5 (f), the equation of the Associated Quadric, at Φ 8 (f), the Triple
Point, one gets a scalar:
∆(f)= Φ 5 (f)(Φ 8 (f)). (2.24)

This object ∆ is a homogeneous covariant of degree 24 and type C. Otherwise stated,
this is a homogeneous invariant for SL(3, C) × SL(4, C). One checks by direct
computation that it is irreducible. From this and the existence of a dense orbit, it is
not difficult to deduce that ∆ is essentially the only invariant. This means that ∆
generates the algebra of the invariants under SL(3, C) × SL(4, C).

Union of the tropes.

Set
Φ 9 = Φ 4 + Φ . (2.25)
2
5
This is a covariant of degree 12 and type Pol (C ), and thus Φ 9 (f) represents some
4
4
quartic surface in CP . One checks that this surface is the union of the four tropes.
3
Trihedron of the double lines.
Remember the classical notion of polar: given an hypersurface of degree d> 1
given by an equation F(z 0 ,...,z r )=0 and a point (Z 0 : ··· : Z r ), the polar of
the hypersurface and the point is the hypersurface of degree d − 1 defined by the
equation
∂F
Z i =0. (2.26)
dz i
i