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
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 (, ). 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 , ). 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 ).
All Enriques surfaces X with ﬁnite Aut(X ) were classiﬁed by S. Kond¯  and
(not constructively) by V. Nikulin . In  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 . 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
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
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