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 Ginvariant 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 [36]. 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 ).
n
0
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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 Fall '04
 IgorDolgachev
 Algebra, Geometry, Vector Space, The Land, Igor V. Dolgachev, Reflection group

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