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 onedimensional representation of Aut(X )
in the space Ω2 (X ) of holomorphic 2forms 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 RiemannRoch 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...
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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

Click to edit the document details