A cremona transformation 61 together with a choice of

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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)...
View Full Document

This note was uploaded on 02/24/2012 for the course MATH 285 taught by Professor Igordolgachev during the Fall '04 term at University of Michigan-Dearborn.

Ask a homework question - tutors are online