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Unformatted text preview: x2 must have the ﬁrst row equal to (1, 0, . . . , 0). Thus the remaining
0
1
N
rows and columns deﬁne an orthogonal matrix of the quadratic form x2 + . . . + x2
1
N
with integer entries. This implies that after reordering the columns and the rows
we get a matrix with ±1 at the diagonal and zero elsewhere. It remains to use the
fact that the transformation leaves the vector kN invariant to conclude that the
matrix is the identity. Thus the transition matrix is the product of the matrices
corresponding to reﬂections in vectors αi , i = 1, . . . , N .
Let us prove the last statement. Suppose g ∗ is the identity on SX . Then
∗
2
g (eN ) = [g −1 (EN )] = eN . Since EN < 0, it is easy to see that EN is homolo−1
gous to g (EN ) only if g (EN ) = EN . This implies that g descends to the surface
XN −1 . Replacing X with XN −1 and repeating the argument, we see that g descends
to XN −2 . Continuing in this way we obtain that g ∗ is the identity and descends
to a projective automorphism g of P2 . If all curves Ei are irreducible, their images
on P2 form an ordered set of N distinct points which must be preserved under g .
Since a square matrix of size 3 × 3 has at most 3 linear independent eigenvectors,
we see that g , and hence g , must be the identity. The general case requires a few
more techniques to prove, and we omit the proof.
The main problem is to describe all possible subgroups of the Weyl group W (EN )
which can be realized as the image of a group G of automorphisms of a rational
surface obtained by blowingup N points in the plane.
First of all we may restrict ourselves to minimal pairs (X, G ⊂ Aut(X )). Minimal
means that any Gequivariant birational regular map f : X → X of rational
surfaces must be an isomorphism. A factorization (5.4) of π : X → P2 can be
extended to a factorization π : X → P2 in such a way that the geometric basis
φπ (a) of SX can be extended to a geometric basis φπ (a) of SX . This gives a
natural inclusion WX ⊂ WX such that the image of G in WX coincides with the
image of G in WX .
The next result goes back to the classical work of S. Kantor [63] and now easily
follows from an equivariant version of Mori’s theory of minimal models.
Theorem 5.3. Let G be a ﬁnite group of automorphisms of a rational surface X
making a minimal pair (X, G). Then either X ∼ P2 , or X is a conic bundle with
=
(SX )G ∼ Z2 or Z, or X is a Del Pezzo surface with (SX )G ∼ Z.
=
=
Here a Del Pezzo surface is a rational surface X with ample −KX .8 Each Del
Pezzo surface is isomorphic to either P2 or P1 × P1 , or it admits a factorization
(5.4) with N ≤ 8 and the images of the points x1 , . . . , xN in P2 are all distinct and
satisfy the following:
• no three are on a line,
• no six are on a conic,
• not all are contained on a plane cubic with one of them being its singular
point (N = 8).
8 This means that in some projective embedding of X its positive multiple is equal to the
fundamental class of a hyperplane section. REFLECTION GRO...
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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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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