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D.6.18 surfacesignature_lib
- Library:
- surfacesignature.lib
- Purpose:
- signature of surface singularity
- Authors:
- Gerhard Pfister pfister@mathematik.uni-kl.de
Muhammad Ahsan Banyamin ahsanbanyamin@gmail.com
Stefan Steidel steidel@mathematik.uni-kl.de
- Overview:
- A library for computing the signature of irreducible surface singularity.
The signature of a surface singularity is defined in [3]. The algorithm we
use has been proposed in [9].
Let g in C[x,y] define an isolated curve singularity at 0 in C^2 and
f:=z^N+g(x,y). The zero-set V:=V(f) in C^3 of f has an isolated singularity
at 0. For a small e>0 let V_e:=V(f-e) in C^3 be the Milnor fibre of (V,0) and
s: H_2(V_e,R) x H_2(V_e,R) ---> R be the intersection form (cf. [1],[7]).
H_2(V_e,R) is an m-dimensional R-vector space, m the Milnor number of (V,0)
(cf. [1],[4],[5],[6]), and s is a symmetric bilinear form.
Let sigma(f) be the signature of s, called the signature of the surface
singularity (V,0). Formulaes to compute the signature are given by Nemethi
(cf. [8],[9]) and van Doorn, Steenbrink (cf. [2]).
We have implemented three approaches using Puiseux expansions, the resolution
of singularities resp. the spectral pairs of the singularity.
- References:
- [1] Arnold, V.I.; Gusein-Zade, S.M.; Varchenko, A.N.: Singularities of
Differentiable Mappings. Vol. 1,2, Birkh"auser (1988).
[2] van Doorn, M.G.M.; Steenbrink, J.H.M.: A supplement to the monodromy
theorem. Abh. Math. Sem. Univ. Hamburg 59, 225-233 (1989).
[3] Durfee, A.H.: The Signature of Smoothings of Complex Surface
Singularities. Mathematische Annalen 232, 85-98 (1978).
[4] de Jong, T.; Pfister, G.: Local Analytic Geometry. Vieweg (2000).
[5] Kerner, D.; Nemethi, A.: The Milnor fibre signature is not semi-continous.
arXiv:0907.5252 (2009).
[6] Kulikov, V.S.: Mixed Hodge Structures and Singularities. Cambridge Tracts
in Mathematics 132, Cambridge University Press (1998).
[7] Nemethi, A.: The real Seifert form and the spectral pairs of isolated
hypersurface singularities. Compositio Mathematica 98, 23-41 (1995).
[8] Nemethi, A.: Dedekind sums and the signature of f(x,y)+z^N. Selecta
Mathematica, New series, Vol. 4, 361-376 (1998).
[9] Nemethi, A.: The Signature of f(x,y)+z^$. Proceedings of Real and Complex
Singularities (C.T.C. Wall's 60th birthday meeting, Liverpool (England),
August 1996), London Math. Soc. Lecture Notes Series 263, 131--149 (1999).
Procedures:
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