Unformatted text preview: s gs gs ) = (gs gs gs ) · · · (gs gs gs )}.
m(s,s )−2 m(s,s )−2 If we impose additional relations
= 1, s ∈ S , we get the deﬁnition of (W, S ).
This deﬁnes an extension of groups
˜
(8.9)
1 → BW → BW → W → 1,
2
˜
where BW is the normal subgroup of BW generated by conjugates of gs .
2
gs REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 47 Brieskorn also proves the following.
Theorem 8.9. Let ∆ ⊂ Cµ be the discriminant of a simple surface singularity.
Then π1 (Cµ \ ∆, u0 ) is isomorphic to the braid group BΓ of the monodromy group
˜
Γ. The regular covering U → ∆ ⊂ Cµ corresponding to the normal subgroup BΓ can
∼ Cµ , where V is a complex
be Γequivariantly extended to the covering V → V /Γ =
vector space of dimension µ on which Γ acts as a reﬂection group. The preimage
of ∆ in V is the union of reﬂection hyperplanes.
Example 8.10. The following properties are equivalent.
• µ+ = 0, µ0 > 0;
• µ+ = 0, µ0 = 2;
• the exceptional curve of a minimal resolution is a nonsingular elliptic curve;
• (V, 0) can be represented by the zero level of one of the polynomials given
in Table 4.
These singularities are called simple elliptic singularities.
Table 4. Simple elliptic surface singularities
Type
P8
X9
J10 Polynomial
3
3
4
z1 + z2 + z3 + λz1 z2 z3
2
4
4
z1 + z2 + z3 + λz1 z2 z3
2
3
6
z1 + z2 + z3 + λz1 z2 z3 Weights Degree
(1, 1, 1)
3
(2, 1, 1)
4
(3, 2, 1)
6 Here the subscript is equal to µ.
Theorem 8.11 (A. Gabrielov [45]). Let M be the Milnor lattice of a simple elliptic
singularity. Then M ⊥ is of rank 2 and M/M ⊥ is isomorphic to the root lattice of
¯
type Eµ−2 . The image G of the monodromy group Γ in (M/M ⊥ ) is the ﬁnite
reﬂection group of type Eµ−2 . The monodromy group is isomorphic to the semidirect product (M ⊥ ⊗ M/M ⊥ ) W (Eµ−2 ) and can be naturally identiﬁed with an
aﬃne complex crystallographic reﬂection group with linear part W (Eµ−2 ).
There is a generalization of Theorem 8.9 to the case of simple elliptic singularities
due to E. Looijenga [74] and [91]. It involves aﬃne crystallographic reﬂection groups
and uses Theorem 3.3.
Example 8.12. The following properties are equivalent:
• µ+ = 1;
• µ+ = 1, µ0 = 1;
• V can be given by equation P (z1 , z2 , z3 ) = 0, where
111
a
b
c
+ + < 1, λ = 0.
P = z1 + z2 + z3 + λz1 z2 z3 ,
a
b
c
These singularities are called hyperbolic unimodal singularities.
Theorem 8.13 (A. Gabrielov [45]). The Milnor lattice M of a hyperbolic singularity is isomorphic to the lattice
Ep,q,r (−1) ⊥ 0 .
The monodromy group is the semidirect product Zµ
O(M/M ⊥ ) is the reﬂection group W (p, q, r ). W (p, q, r ). Its image in 48 IGOR V. DOLGACHEV The previous classes of isolated surface singularities are characterized by the
condition µ+ ≤ 1. If µ+ ≥ 2, the monodromy group is always of ﬁnite index in
O(M ) (see [40], [41]). Together with the previous theorems this implies that the
monodromy group...
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 Fall '04
 IgorDolgachev
 Algebra, Geometry, Vector Space, The Land, Igor V. Dolgachev, Reflection group

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