Theorem 63 let x pq1 p1 consider the natural diagonal

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Unformatted text preview: izations (5.4) for π and σ is called a marked Cremona transformation. It follows from the proof of Theorem 5.2 that any marked Cremona transformation defines an element of the Coxeter group W (EN ). The corresponding matrix is called the characteristic matrix of the marked Cremona transformation. Example 6.1. Let T be the standard quadratic Cremona transformation defined by T : (x0 , x1 , x2 ) → (x−1 , x−1 , x−1 ) 0 1 2 (to make sense of this one has to multiply all coordinates at the output by x0 x1 x2 ). It is not defined at points p1 = (1, 0, 0), p2 = (0, 1, 0), p3 = (0, 0, 1). Let π : X = X3 → X2 → X1 → X0 = P2 be the composition of the blow-up of p1 , then the blow-up of the pre-image of p2 , and finally the pre-image of p3 . It is easy to see on the open subset U where T is defined the coordinate line ti = 0 is mapped to the point pi . This implies that the Zariski closures on X of the pre-images of REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 37 intersections of these lines with U are the curves E1 , E2 , E3 on X which are blown down to the points p1 , p2 , p3 under T . The factorization for σ : X → P2 could be chosen in such a way that Ei = Ei are the exceptional curves. These curves define a new geometric basis in X with e1 = e0 − e2 − e3 , e1 = e0 − e1 − e3 , e1 = e0 − e0 − e1 . Since kX = −3e0 + e1 + e2 = e3 = −3e0 + e1 + e2 + e3 , we also get e0 = 2e0 − e1 − e2 − e3 . The corresponding transformation is the reflection with respect to the vector e0 − e1 − e2 − e3 . Given any set of 3 noncollinear points q1 , q2 , q3 , one can find a quadratic9 Cremona transformation T with indeterminacy points q1 , q2 , q3 . For this we choose a projective transformation g which sends qi to pi and take T = T ◦ g. Theorem 6.2. Let A be the matrix representing an element from W (EN ). Then there exists a marked Cremona transformation whose characteristic matrix is equal to A. Proof. Let A be the matrix of w ∈ W (EN ) with respect to the standard basis e0 , . . . , eN of I1,N . Its first column is a vector (m0 , −m1 , . . . , −mN ). Write w as a word in reflections rai and use induction on the length of w to prove the Noether inequalities mi ≥ 0, i ≥ 1 and m0 > 3 max{mi , i ≥ 1}. Now, let us use induction on the length of w to show that the linear system |OP2 (m0 ) − m1 p1 − . . . − mN pN )| is homaloidal for some points p1 , . . . , pN in general position. If w is a simple reflection rai we get w(e0 ) = e0 or 2e0 − e1 − e2 − e3 . In the first case the linear system defines a projective transformation, and in the second case it defines a standard quadratic transformation if we choose three noncollinear points p1 , p2 , p3 . Now write w = rai w where the length of w is less than the length of w. By induction, w defines a homaloidal linear system |OP2 (m0 ) − m1 p1 − . . . − mN pN )| for some points p1 , . . . , pN in general position. Let Φ : P2 − → P2 be the corresponding Cremona transformation. If i = 0, the reflection permutes the mi , i > 0, an...
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