A cremona transformation 61 together with a choice of

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Unformatted text preview: rs(H 2 (X, Z)) = Z/2Z. The cover Y → X corresponding to the generator of H1 (X, Z) is a K3 surface. So, Enriques surfaces correspond to pairs (Y, τ ), where Y is a K3 surface and τ is its fixed-point-free involution. We have SX (−1) = U ⊥ E8 ∼ E10 . = The reflection group of E10 is the group W (2, 3, 7). Since its fundamental polytope does not have nontrivial symmetries, we obtain from (4.5) (5.12) O(SX )+ = Ref(SX (−1)) ∼ W (2, 3, 7). = + Let WX be the subgroup of O(SX ) generated by reflections rα , where α is the image in SX of the cohomology class of a (−2)-curve on X . The following theorem follows (but nontrivially) from the Global Torelli Theorem for K3 surfaces ([83], [84]). Theorem 5.12. Let AX be the image of Aut(X ) in O(SX ). Then AX ⊂ O(SX )+ , + + its intersection with WX is trivial and WX AX is of finite index in O(SX ). + This gives as a corollary that Aut(X ) is finite if and only if WX is of finite index in O(SX ). A general Enriques surface (in some precise meaning) does not contain (−2)-curves, so A(X ) is isomorphic to a subgroup of finite index of the orthogonal group O(E10 ). In fact, more precisely, the group Aut(X ) is isomorphic to the 2level congruence subgroup of W (E10 ) defined in (5.10) (see [8], [84]). The fact that the automorphism group of a general Enriques surface and a general Coble surface are isomorphic is not a coincidence, but I am not going to explain it here (see [31]). All Enriques surfaces X with finite Aut(X ) were classified by S. Kond¯ [68] and o (not constructively) by V. Nikulin [85]. In [30] I gave an example of an Enriques surface with finite automorphism group, believing that it was the first example of this kind. After the paper had been published I found that the existence of another example was claimed much earlier by G. Fano [43]. However, his arguments are very obscure and impossible to follow. The Coxeter diagram of the reflection group + WX in my example is given in Figure 13. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 35 † • hhhhh ††††www wwI qqq q •q •I II  I@   @@ "" @@ "" "@ •@ • • • • • @@ "" @I "" II " II   w • www • q w††† qqq ††hhhhhq • Figure 13. 6. Cremona transformations 6.1. Plane Cremona transformations. A Cremona transformation of projective space Pn (as always over complex numbers) is a birational map of algebraic varieties. It can be given in projective coordinates by n + 1 homogeneous polynomials of the same degree d: (6.1) T : (t0 , . . . , tn ) → (P0 (t0 , . . . , tn ), . . . , Pn (t0 , . . . , tn )). Dividing by a common multiple of the polynomials we may assume that the map is not defined on a closed subset of codimension ≥ 2, the set of common zeros of the polynomials P0 , . . . , Pn . Let U = dom(T ) be the largest open subset where T is defined and let X be the Zariski closure of the graph of T : U → Pn in Pn × Pn . By considering the two projections of X , we get a commutative diagram of birational maps: (6.2)...
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