# Theorem 63 let x pq1 p1 consider the natural diagonal

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

This is the end of the preview. Sign up to access the rest of the document.

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 &gt; 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 &gt; 0, an...
View Full Document

Ask a homework question - tutors are online