# A divisor class x is called nef if x d 0 for any

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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 counter-example 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 bi-degree (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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