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
1 → BW → BW → W → 1,
where BW is the normal subgroup of BW generated by conjugates of gs .
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 Γ-equivariantly extended to the covering V → V /Γ =
vector space of dimension µ on which Γ acts as a reﬂection group. The pre-image
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
z1 + z2 + z3 + λz1 z2 z3
z1 + z2 + z3 + λz1 z2 z3
z1 + z2 + z3 + λz1 z2 z3 Weights Degree
(1, 1, 1)
(2, 1, 1)
(3, 2, 1)
6 Here the subscript is equal to µ.
Theorem 8.11 (A. Gabrielov ). 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  and . 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
+ + < 1, λ = 0.
P = z1 + z2 + z3 + λz1 z2 z3 ,
These singularities are called hyperbolic unimodal singularities.
Theorem 8.13 (A. Gabrielov ). The Milnor lattice M of a hyperbolic singularity is isomorphic to the lattice
Ep,q,r (−1) ⊥ 0 .
The monodromy group is the semi-direct 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 , ). Together with the previous theorems this implies that the
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