On Moduli Spaces of Semiquasihomogeneous Singularities

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


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
$${\tau _i}\left( A \right): = {\dim _\mathbb{C}}\mathbb{C}\left[\kern-0.15em\left[ {{x_1}, \ldots ,{x_n}} \right]\kern-0.15em\right]/\left( {f,\frac{{\partial f}}{{\partial {x_1}}}, \ldots ,\frac{{\partial f}}{{\partial {x_n}}},{H^i}} \right).$$


Principal Part Hilbert Function Integral Manifold Weighted Projective Space Coarse Modulus Space 
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© Birkhäuser Verlag Basel/Switzerland 1996

Authors and Affiliations

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

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