Unformatted text preview: s γi generate π1 (C \ {t1 , . . . , tµ }; t0 ), the map
s∗ : π1 (C \ {t1 , . . . , tµ }; t0 ) → π1 (U \ ∆; u0 )
is surjective and the images gi of [γi ] under the composition of the monodromy map
and s∗ generate the monodromy group Γ. Applying the PicardLefschetz formula
(8.5) we obtain, for any x ∈ M ,
gi (x) = x − (−1)n(n−1)/2 (x, δi )δi , i = 1, . . . , µ. This shows that Γ is generated by µ reﬂections in elements δi satisfying (8.4). This
proves the claim.
Remark 8.3. It is known that for any isolated hypersurface singularity given by a
weighted homogeneous polynomial P with isolated critical point at 0 the Milnor
n
lattice is isomorphic to Hc (P −1 ( ), Z). In this case we can deﬁne the monodromy
group as the image of the monodromy map
n
π1 (Cµ \ ∆; u0 ) → O(Hc (π −1 (u0 ), Z)), where u0 ∈ Cµ \ ∆. REFLECTION GROUPS IN ALGEBRAIC GEOMETRY 45 One can relate the global monodromy group with the classical monodromy operator.
Theorem 8.4. Let T be a classical monodromy operator and s1 , . . . , sµ be the
reﬂections in O(M ) corresponding to a choice of paths γi as above. Then ±T is
conjugate to the product s1 · · · sµ .
The product s1 · · · sµ is an analog of a Coxeter element in the reﬂection group
of M (see Example 5.7). However, the Weyl group of M is not a Coxeter group in
general.
8.2. Surface singularities. Assume now that n = 2. We will represent a surface
singularity by an isolated singular point 0 of an aﬃne surface X ⊂ C3 .
Let π : X → X be a resolution of the singularity (X, 0). This means that X
is a nonsingular algebraic surface and π is a proper holomorphic map which is an
isomorphism over U = X \ {0}. We may assume it to be minimal, i.e. does not
contain (−1)curves in its ﬁbre over 0. This assumption makes it unique up to
isomorphism.
One deﬁnes the following invariants of (X, 0). The ﬁrst invariant is the genus δpa
of (X, 0). This is the dimension of the ﬁrst cohomology space of X with coeﬃcient
in the structure sheaf OX of regular (or holomorphic) functions on X . In spite of
X not being compact, this space is ﬁnitedimensional.
Our second invariant is the canonical class square δK 2 . To deﬁne it we use that
there is a rational diﬀerential 2form on X which has no poles or zeros on X \ {0}.
Its extension to X has divisor D supported at the exceptional ﬁbre. It represents
2
an element [D] in Hc (X , Z). Using the cupproduct pairing we get the number
2
[D] , which we take for δK 2 .
Finally we consider the ﬁbre E = π −1 (0) of a minimal resolution. This is a
(usually reducible) holomorphic curve on X . We denote its Betti numbers by bi .
For example, b2 is the number of irreducible components of E .
Since X is deﬁned uniquely up to isomorphism, the numbers δpa , δK 2 , bi are
welldeﬁned.
Remark 8.5. The notations δpa and δK 2 are explained as follows. Suppose Y is
a projective surface of degree d in P3 with isolated singularities y1 , . . . , yk . Let Y
be its minimal resolution. Deﬁne the arithmetic gen...
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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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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