Gizatullin but a counter example was found by b

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Unformatted text preview: UPS IN ALGEBRAIC GEOMETRY 29 A conic bundle is a rational surface which admits a regular map to a nonsingular curve with fibres isomorphic to a conic (nonsingular or the union of two distinct lines). A partial classification of finite groups G which can be realized as groups of automorphisms of some rational surface was given by S. Kantor (for a complete classification and the history of the problem see [33]). Example 5.4. Let X be a Del Pezzo surface with N + 1 = rank SX . We refer to the number 9 − N as the degree of X . A Del Pezzo surface of degree d > 2 is isomorphic to a nonsingular surface of degree d in Pd . The most famous example is a cubic surface in P3 with 27 lines on it. The Weyl group W (E6 ) is isomorphic to the group of 27 lines on a cubic surface, i.e. the subgroup of the permutation group Σ27 which preserves the incidence relation between the lines. Although the group of automorphisms of a general cubic surface is trivial, some special cubic surfaces admit nontrivial finite automorphism groups. All of them were essentially classified in the 19th century. The situation with infinite groups is more interesting and difficult. Since EN is negative definite for N ≤ 8, a basic rational surface X with infinite automorphism group is obtained by blowing up N ≥ 9 points. It is known that when the points are in general position, in some precisely defined sense, the group Aut(X ) is trivial [57], [67]. So surfaces with nontrivial automorphisms are obtained by blowing up a set of points in some special position. Example 5.5. Let X be obtained by blowing up 9 points x1 , . . . , x9 contained in two distinct irreducible plane cubic curves F, G. The surface admits a fibration X → P1 with general fibre an elliptic curve. The image of each fibre in P2 is a plane cubic from the pencil of cubics spanned by the curves F, G. The exceptional curves E1 , . . . , E9 are sections of this fibration. Fix one of them, say E1 , and equip each nonsingular fibre Xt with the group law with the zero point Xt ∩ E1 . Take a point x ∈ Xt and consider the sum x + pi (t), where pi (t) = Xt ∩ Ei . This defines an automorphism on an open subset of X which can be extended to an automorphism gi of X . When the points x1 , . . . , x9 are general enough, the automorphisms g2 , . . . , g9 generate a free abelian group of rank 8. In general case it is a finitely generated group of rank ≤ 8. In the representation of Aut(X ) in WX ∼ W (E9 ) the = image of this group is the lattice subgroup of the euclidean reflection group of type E8 . This example can be generalized by taking general points x1 , . . . , x9 with the property that there exists an irreducible curve of degree 3m such that each xi is its singular point of multiplicity m. In this case the image of Aut(X ) in W (E9 ) is the subgroup of the lattice subgroup Z8 such that the quotient group is isomorphic to (Z/mZ)8 (see [36], [46]). Example 5.6. Let X be obtained by blowing up 10 points x1 , . . . , x10 with the property that there exists an irr...
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