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# Since the points are in general position we may

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Unformatted text preview: Xe ee }} σ } ee }} ee } ~} T   / 2 P2     P π By taking a resolution of singularities X → X , we may assume that such a diagram exists with X nonsingular. We call it a resolution of indeterminacy points of T . We denote the projective space of homogeneous polynomials of degree d in n + 1 variables by |OPn (d)|. The projective subspace of dimension n spanned by the polynomials P0 , . . . , Pn deﬁning the map T is denoted by L(T ) and is called the linear system deﬁning T . It depends only on T (recall that two birational maps are equal if they coincide on a Zariski open subset). Any n-dimensional projective subspace L of |OPn (d)| (linear system) which consists of polynomials without a common factor deﬁnes a rational map Pn − → Pn by simply choosing a linear independent ordered set of n + 1 polynomials from L. When the map happens to be birational, the linear system is called homaloidal. One obtains L(T ) as follows. First one considers the linear system |OPn (1)| of hyperplanes in the target Pn . The pre-image of its member on X under the map σ is a hypersurface on X ; we push it down by π and get a hypersurface on the domain Pn . The set of such hypersurfaces forms a linear system L(T ). 36 IGOR V. DOLGACHEV Now let us assume that n = 2. As in section 5.2 we consider a factorization (5.4) of π , (6.3) π N −1 π π π N 2 1 π : X = XN −→ XN −1 −→ . . . −→ X1 −→ X0 = P2 , where πk+1 is the blow-up of a point xk ∈ Xk , k = 0, . . . , N − 1. Recall that the multiplicity multx (D) of a hypersurface D at a point x on a nonsingular variety is the degree of the ﬁrst nonzero homogeneous part in the Taylor expansion of its local equation at x. Deﬁne inductively the numbers mi as follows. Let L = L(T ) be the linear system of curves on P2 deﬁning T . First we set m1 = min multx1 D. D ∈L ∗ π1 (L) on X1 which consists of the full pre-images of hypersurfaces The linear system from L(T ) on X1 has the hypersurface m1 E1 as a ﬁxed component. Let ∗ L1 = π1 (L) − m1 E1 . This is a linear system on X1 without ﬁxed components. Suppose m1 , . . . , mi and L1 , . . . , Li have been deﬁned. Then we set mi+1 = min multxi+1 D, D ∈Li ∗ Li+1 = πi+1 (Li ) − mi Ei+1 . It follows from the deﬁnition that N LN = π ∗ (L) − mi E i i=1 has no ﬁxed components and is equal to the pre-image of |OP2 (1)| under σ . The image of LN in P2 is equal to the linear system L(T ). It is denoted by |OP2 (d) − m1 x1 − . . . − mN xN |. The meaning of the notation is that L(T ) consists of plane curves of degree d which pass through the points xi with multiplicities ≥ mi . We have similar decomposition for the map σ which deﬁnes the linear system L(T −1 ). As we have shown in section 5.2, two factorizations of birational regular maps from X to P2 in a sequence of blow-ups deﬁne two geometric bases of the lattice SX . A Cremona transformation (6.1) together with a choice of a diagram (6.2) and the factor...
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