Its orthogonal complement is a lattice of rank 22 12

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Unformatted text preview: nonzero holomorphic vector fields and hence the Lie algebra of the maximal Lie subgroup of Aut(X ) is trivial. This shows that the group Aut(X ) is a discrete topological group and the kernel of the natural representation (5.2) of Aut(X ) in O(SX ) is a finite group. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 31 Remark 5.9. One can say more about the kernel H of homomorphism (5.2) (see [84], §10). Let χ : Aut(X ) → C∗ be the one-dimensional representation of Aut(X ) in the space Ω2 (X ) of holomorphic 2-forms on X . The image of χ is a cyclic group of some order n. First Nikulin proves that the value of the Euler function φ(n) divides 22 − rankSX . Next he proves that the restriction of χ to H is injective. This implies that H is a cyclic group of order dividing n. All possible values of n which can occur are known (see [69]). The largest one is equal to 66 and can be realized for a K3 surface birationally isomorphic to a surface in the weighted projective space P(1, 6, 22, 33) given by the equation x66 + y 11 + z 3 + w2 = 0. It follows from the adjunction formula that any smooth rational curve on a K3 + surface is a (−2)-curve. The class of this curve in SX defines a reflection. Let WX denote the subgroup of O(SX ) generated by these reflections. Proposition 5.10. + Ref−2 (SX ) = WX . Proof. Let C be an irreducible curve on X . By the adjunction formula, [C ]2 ≥ −2 and C 2 = −2 if and only if C is a (−2)-curve. We call a divisor class effective if it can be represented by a (possibly reducible) algebraic curve on X . Using the Riemann-Roch Theorem, one shows that any divisor class x with x2 ≥ −2 is either effective or its negative is effective. Let D = i∈I Ci be any algebraic curve written as a sum of its irreducible components. We have [D] · [C ] ≥ 0 for any irreducible curve C unless C is an irreducible component of D with C 2 = −2. A divisor class x is called nef if x · d ≥ 0 for any effective divisor class d. It is not + difficult to show that the WX -orbit of any effective divisor x with x2 ≥ 0 contains a unique nef divisor (use that x · e < 0 for some effective e with e2 = −2 implying 0 re (x) · e > 0). Let VX = {x ∈ VX : x2 > 0}. We take for the model of the hyperbolic n 0 space H associated with SX the connected component of VX /R+ ⊂ P(VX ) which 0 contains the images of effective divisors from VX . Then the image P + in H n of 0 the convex hull N of nef effective divisors from VX is a fundamental polytope for + the reflection group WX . Its fundamental roots are the classes of (−2)-curves. Let rα be a reflection from Ref−2 (SX ). Replacing α with −α we may assume that α is effective. Suppose α is a fundamental root for a fundamental polytope P of Ref−2 (SX ) which is contained in P + . Since all vectors from N satisfy x · α ≥ 0, + we see that P + ⊂ P and hence P = P + . This shows that WX and Ref−2 (SX ) are defined by the same convex polytope and hence the groups are equal. The following result follows from the fundamental Gl...
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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.

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