Discrete Holomorphic Local Dynamical Systems

90 Eric Bedford
2.4 Three Involutions
Let us apply the preceding discussion to three birational maps that are involutions.
These are quadratic maps, presented in the order of increasing degeneracy. All three
can be turned into regular involutions after blowing up. The degree of degeneracy
will correspond to the depth of blowup that is necessary to remove the exceptional
curves and indeterminate points. The linear fractional recurrences, discussed in the
following section, are of the form A ◦ J, whereA is linear, and J is the Cremona
inversion, our first involution. We note, too, that the general quadratic Hénon map
is also given by composing the third involution L with an affine map:
(y,y 2 + c − δx)=L ◦ A, A(x,y)=(y,−δx + −c).
These involutions play a basic role in the classification of the Cremona group of birational
maps of the plane. Namely, if f is a quadratic, birational map of the plane,
then f is linearly conjugate to a map of the form A ◦ φ, whereφ is one of the involutions
studied in this section (see [CD]). The analogous classification of the cubic
Cremona transformations, which is more complicated, is also given in [CD].
The first of these involutions is the Cremona inversion J of P 2 which is defined
by
J[x0 : x1 : x2]=[J0 : J1 : J2]=[x −1
0 : x −1
1 : x −1
2 ]=[x1x2 : x0x2 : x0x1].
It is evident that J = J −1 .Wesetp0 =[1:0:0], p1 =[0:1:0],andp2 =[0:0:1],
and let L j, 0≤ j ≤ 2, denote the side of the triangle p0p1p2 which is opposite p j.
We let X denote the manifold obtained by blowing up P 2 at the points p j,0≤ j ≤ 2.
Figure 7 shows that we may visualize this as an inversion in a triangle sending p j ↔
Σ j for j = 0,1,2.
Exercise: Let JX : X → X. Show that JX is holomorphic and maps L j ↔ Pj for
0 ≤ j ≤ 2.
In H 2 (X), wehaveL0 = H − P1 − P2, etc. Specifically, if we pull back a line
∑ajx j = 0, then we have ∑ajJj = 0, which is a quadric which contains all three
points p j, 0≤ j ≤ 2. Thus we see that J ∗ H = 2H on H 2 (P 2 ),and
Σ 1
p 0
p 2
Σ 2
J ∗ X HX = 2HX − P0 − P1 − P2 ∈ H 2 (X).
Σ 0
Fig. 7 Inversion in a triangle
p 1
p 0
Σ 1
p 2
Σ 2
Σ 0
p 1