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Unformatted text preview: educible curve of degree 6 with ordinary double
points at each xi (a Coble surface ). The image of Aut(X ) in WX ∼ W (E10 ) is the
=
subgroup
(5.10) W (E10 )(2) = {g ∈ W (E10 ) : g (x) − x ∈ 2E10 for all x ∈ E10 } (see [20]). This group is the smallest normal subgroup which contains the involution
of the lattice E10 = U ⊥ E8 equal to (idU , −idE8 ). 30 IGOR V. DOLGACHEV Example 5.7. Let (W, S ) be a Coxeter system with ﬁnite set S . A Coxeter element
of (W, S ) is the product of elements of S taken in some order. Its conjugacy class
does not depend on the order if the Coxeter diagram is a tree ([14], Chapter V,
§6, Lemma 1). The order of a Coxeter element is ﬁnite if and only if W is ﬁnite.
Let hN be a Coxeter element of W = W (EN ). In a recent paper [78] C. McMullen
realizes hN by an automorphism of a rational surface. The corresponding surface
X is obtained by blowing up N points in special position lying on a cuspidal plane
cubic curve. One can check that for N = 9 or 10 a Coxeter element does not
belong to the subgroup described in the previous two examples. It is not known
whether a rational surface realizing a Coxeter element is unique up to isomorphism
for N ≥ 9. It is known to be unique for N ≤ 8. For example, for N = 6 the surface
is isomorphic to the cubic surface
3
3
2
2
T0 + T1 + T3 T1 + T2 T3 = 0. The order of h6 is equal to 12.
Until very recently, all known examples of minimal pairs (X, G) with inﬁnite G
satisﬁed the following condition:
• There exists m > 0 such that the linear system  − mKX  is not empty or,
in another words, the cohomology class mc1 (X ) can be represented by an
algebraic curve.
Without the minimality condition the necessity of this condition was conjectured
by M. Gizatullin, but a counterexample was found by B. Harbourne [54]. A recent
preprint of Eric Bedford and Kyounghee Kim [10] contains an example of a minimal
surface with inﬁnite automorphism group with  − mKX  = ∅ for all m > 0.
5.3. K3 surfaces. These are surfaces from case 2 (i) of Theorem 5.1. They are
characterized by the conditions
c1 (X ) = 0, H 1 (X, C) = 0. In fact, all K3 surfaces are simply connected and belong to the same diﬀeomorphism
type.
Examples 5.8. 1) X is a nonsingular surface of degree 4 in P3 .
2) X is the double cover of a rational surface Y branched along a nonsingular
curve W whose cohomology class [W ] is equal to −2KX . For example, one may
take Y = P2 and W a nonsingular curve of degree 6. Or, one takes Y = P1 × P1
and W a curve of bidegree (4, 4).
3) Let A be a compact complex torus which happens to be a projective algebraic
variety. This is a surface from case 2 (iii). The involution τ : a → −a has 16 ﬁxed
points, and the orbit space A/(τ ) acquires 16 ordinary double points. A minimal
nonsingular surface birationally equivalent to the quotient is a K3 surface, called
the Kummer surface associated to A.
Let Aut(X ) be the group of biregular automorphisms of X . It is known that X
does not admit...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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