Then m is of rank 2 and mm is isomorphic to the root

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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 Picard-Lefschetz 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 finite-dimensional. Our second invariant is the canonical class square δK 2 . To define it we use that there is a rational differential 2-form 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 cup-product 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 well-defined. 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 Michigan-Dearborn.

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