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On Moduli Spaces of Semiquasihomogeneous Singularities

Conference paper
Part of the Progress in Mathematics book series (PM, volume 134)

Abstract

Let \(A = \mathbb{C}\left[\kern-0.15em\left[ {{{x}_{1}}, \ldots {{x}_{n}}} \right]\kern-0.15em\right]/\left( f \right) \) be the complete local ring of a hypersurface singularity. A is called semiquasihomogeneous with weights w 1,…,w n if f = f 0 + f 1, f 0 a quasihomogeneous polynomial defining an isolated singularity, and deg f 0 < deg f 1. We assume that w 1,…,w n are positive integers and let deg always denote the weighted degree, i.e., deg X α = w 1α1 +⋯+ w n α n for a monomial \({X^\alpha } = X_1^{{\alpha _1}} \ldots X_1^{{\alpha _n}}\). For an arbitrary power series f, deg f denotes the smallest weighted degree of a monomial occurring in f. By definition, all monomials of a quasihomogeneous polynomial have the same degree. The singularity with local ring A 0 = ℂ〚x 1,…,x n 〛/(f 0) is called the principal part of A. If the moduli stratum of A 0 has dimension 0, i.e., the τ-constant stratum in the semiuniversal deformation of A 0 is a reduced point, then A 0 is uniquely determined by the weights. Let H i = H i (ℂ〚x 1,…,x n 〛) be the ideal generated by all quasihomogeneous polynomials of degree ≥ iw, w:= min{w 1,…,w n }. This (weighted) degree-filtration defines a Hilbert-function τ on the Tjurina algebra of A by
$${\tau _i}\left( A \right): = {\dim _\mathbb{C}}\mathbb{C}\left[\kern-0.15em\left[ {{x_1}, \ldots ,{x_n}} \right]\kern-0.15em\right]/\left( {f,\frac{{\partial f}}{{\partial {x_1}}}, \ldots ,\frac{{\partial f}}{{\partial {x_n}}},{H^i}} \right).$$

Keywords

Principal Part Hilbert Function Integral Manifold Weighted Projective Space Coarse Modulus Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Birkhäuser Verlag Basel/Switzerland 1996

Authors and Affiliations

  1. 1.Fachbereich MathematikUniversität KaiserslauternKaiserslauternGermany
  2. 2.Fachbereich MathematikHumboldt-Univerität zu BerlinBerlinGermany

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