One can dene the notion of a g equivariant

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Unformatted text preview: us of a nonsingular projective surface V as pa = −q + pg , where q = dim H 1 (V, OV ) = dim Ω1 (X ) is the dimension of the space of holomorphic 1-forms on V and pg = dim H 2 (V, OV ) = dim Ω2 (X ) is the dimension of the space of holomorphic 2-forms on V . Let Fd be a nonsingular hypersurface of degree d in P3 . Then k p a ( Y ) = p a ( Fd ) − k δpa (xi ), 2 2 KY = KFd − i=1 δK 2 (xi ). i=1 2 The numbers pa (Fd ) and KFd are easy to compute. We have pa (Fd ) = pg (Fd ) = (d − 1)(d − 2)(d − 3)/6, 2 KFd = d(d − 4)2 . Theorem 8.6 (J. Steenbrink [108]). Let µ be the Milnor number of a surface singularity (V, 0). Then the signature (µ+ , µ− , µ0 ) of the Milnor lattice is given as follows: µ0 = b1 , µ+ + µ− + µ0 = µ, µ+ − µ− = −δK 2 − b2 − 8δpa . 46 IGOR V. DOLGACHEV Example 8.7. The following properties are equivalent: • δpa = 0; • µ+ = µ0 = 0; • δK 2 = 0; • b2 = µ; • the exceptional curve of a minimal resolution is the union of nodal curves; • V can be given by equation P (z1 , z2 , z3 ) = 0, where P is a weighted homogeneous polynomial of degree d with respect to positive weights q1 , q2 , q3 such that d − q1 − q2 − q3 < 0; • (V, 0) is isomorphic to an affine surface with ring of regular functions isomorphic to the ring of invariant polynomials of a finite subgroup G ⊂ SL(2, C). Table 3 gives the list of isomorphism classes of surface singularities characterized by the previous properties. They go under many different names: simple singularities, ADE singularities, Du Val singularities, double rational points, Gorenstein quotient singularities, Klein singularities. Table 3. Simple surface singularities Type Polynomial 2 A2k , k ≥ 1 Z1 Z2 + Z3 k+1 2 A2k+1 , k ≥ 0 Z1 Z2 + Z3 k+2 n 2 2 Dn , n ≥ 4 Z1 + Z2 Z3 + Z2 −1 2 3 4 E6 Z1 + Z2 + Z3 2 3 3 E7 Z1 + Z2 + Z2 Z3 2 3 5 E8 Z1 + Z2 + Z3 Weights Degree G (2k + 1, 2k + 1, 2) 4k + 2 C4k (k + 1, k + 1, 1) 2k + 2 C2k+1 ¯ (n − 1, 2, n − 2) 2n − 2 D4n−4 ¯ T24 (6, 4, 3) 12 ¯ O48 (9, 6, 4) 18 ¯ I120 (15, 10, 6) 30 The following result follows immediately from Brieskorn’s and Tjurina’s construction of simultaneous resolution of simple surface singularities [15], [92], [111]. Theorem 8.8. The Milnor lattice of a simple surface singularity is a finite root lattice of the type indicated in the first column of Table 3 with quadratic form multiplied by −1. The monodromy group is the corresponding finite real reflection group. It is known that the incidence graph of the irreducible components of a minimal resolution of singularities is the Coxeter graph of the corresponding type. This was first observed by P. Du Val [37]. Consider the monodromy group of a simple surface singularity Γ of (X, 0) as the image of the monodromy map (Remark 8.3). For each Coxeter system (W, S ) with Coxeter matrix (mss ) one defines the associated Artin-Brieskorn braid group BW by presentation BW = {gs , s ∈ S : (gs gs gs ) · · · (g...
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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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