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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
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 ).
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
, . 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
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 , ).
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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