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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 1forms on V and pg = dim H 2 (V, OV ) = dim Ω2 (X ) is
the dimension of the space of holomorphic 2forms 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 ArtinBrieskorn 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 MichiganDearborn.
 Fall '04
 IgorDolgachev
 Algebra, Geometry, The Land

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