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Unformatted text preview: d the linear system is still homaloidal and the set of points does not
change. Since the points are in general position we may assume that p1 , p2 , p3
are distinct noncollinear points. Composing Φ with a projective transformation
we may assume that p1 = (1, 0, 0), p2 = (0, 1, 0), p3 = (0, 0, 1). The computation
from the previous example shows that ra0 w (e0 ) = m0 e0 − i≥1 mi ei , where mi =
2mi − m1 − m2 − m3 , i ≤ 3, and mi = mi , i > 3. Let Φ = Φ ◦ T , where T is the
standard quadratic transformation discussed in the previous example. Then it is
easy to see that Φ is given by the linear system |OP2 (m0 ) − m1 p1 − . . . − mN pN )|.
Let w ∈ W (EN ) correspond to the characteristic matrix of Φ (with respect to an
appropriate marking). We have proved that w(e0 ) = w(e0 ). This implies that
ww−1 (e0 ) = e0 . The matrix representing an element from O(1, N ) whose ﬁrst
column is the unit vector is a diagonal matrix with ±1 at the diagonal. As we have
seen already in the proof of Theorem 5.2, this implies that w = w.
One can apply Theorem 6.2 to list the types (m0 , m1 , . . . , mN ) of all homaloidal
linear systems with N indeterminacy points. They correspond to the orbit of the
vector e0 with respect to the group W (EN ). In particular, the number of types is
ﬁnite only for N ≤ 8.
9 i.e. deﬁned by polynomials of degree 2. 38 IGOR V. DOLGACHEV 6.2. Cremona action of W (p, q, r ). Consider the natural diagonal action of the
group G = PGL(n + 1, C) on (Pn )N , where m = N − n − 2 ≥ 0. A general orbit
contains a unique point set (p1 , . . . , pN ) with the ﬁrst n + 2 points equal to the
set of reference points (1, 0, . . . , 0), . . . , (0, . . . , 0, 1), (1, . . . , 1). This easily implies
that the ﬁeld of G-invariant rational functions on (Pn )N is isomorphic to the ﬁeld of
rational functions on (Pn )m and hence is isomorphic to the ﬁeld of rational functions
C(z1 , . . . , znm ). The symmetric group ΣN acts naturally on this ﬁeld via its action
on (Pn )N by permuting the factors. Assume n ≥ 2 and consider ΣN as a subgroup
W (1, n + 2, m + 1) of the Coxeter group of type W (2, n + 1, m + 1) corresponding
to the subdiagram of type AN −1 of the Coxeter diagram of W (2, n + 1, m + 1). In
1917 A. Coble extended the action of ΣN on the ﬁeld C(z1 , . . . , znm ) to the action
of the whole group W (2, n + 1, m + 1). This construction is explained in modern
terms in . In Coble’s action the remaining generator of the Coxeter group acts
as a standard quadratic transformation Pn − → Pn deﬁned by
T : (x0 , . . . , xn ) → (x−1 , . . . , x−1 ).
One takes a point set (p1 , . . . , pN ), where the ﬁrst n + 2 points are the reference
points, then applies T to the remaining points to get a new set,
(p1 , . . . , pn+1 , T (pn+2 ), . . . , T (pN )).
The Cremona action is the corresponding homomorphism of groups
W (2, n + 1, m + 1) → AutC (C(z1 , . . . , znm )).
One can show that for N ≥ 9, this homomorphism does not arise...
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