For example the monster group f1 is a quotient of w 4

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 [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...
View Full Document

This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

Ask a homework question - tutors are online