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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
ﬁxedpointfree 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
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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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 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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