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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
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
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