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Quasi Ordinary Singularities, Essential Divisors and Poincare Series

Pedro Daniel Gonzalez Perez, Fernando Hernando

Abstract

We define Poincaré series associated to a toric or analytically irreducible quasi-ordinary hypersurface singularity, (S,0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multi-graded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincaré series is a rational function with integer coefficients, which can be defined also as an integral with respect of the Euler characteristic, over the projectivization of the analytic algebra of the singularity, of a function defined by the valuations. In particular, the Poincaré series associated to the set of divisorial valuations associated to the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasi-ordinary hypersurface case we prove that this Poincaré series determines and it is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity.

Quasi Ordinary Singularities, Essential Divisors and Poincare Series

Abstract

We define Poincaré series associated to a toric or analytically irreducible quasi-ordinary hypersurface singularity, (S,0), by a finite sequence of monomial valuations, such that at least one of them is centered at the origin 0. This involves the definition of a multi-graded ring associated to the analytic algebra of the singularity by the sequence of valuations. We prove that the Poincaré series is a rational function with integer coefficients, which can be defined also as an integral with respect of the Euler characteristic, over the projectivization of the analytic algebra of the singularity, of a function defined by the valuations. In particular, the Poincaré series associated to the set of divisorial valuations associated to the essential divisors, considered both over the singular locus and over the point 0, is an analytic invariant of the singularity. In the quasi-ordinary hypersurface case we prove that this Poincaré series determines and it is determined by the normalized sequence of characteristic monomials. These monomials in the analytic case define a complete invariant of the embedded topological type of the hypersurface singularity.

Paper Structure

This paper contains 21 sections, 20 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Poincaré series of singularities
  3. Essential divisors and the Nash map
  4. Quasi-ordinary singularities and equisingularity
  5. The notion of semigroup of a quasi-ordinary hypersurface
  6. The zeta function of geometric monodromy
  7. Structure of the paper and results
  8. A reminder of toric geometry
  9. Quasi-Ordinary Singularities
  10. Characteristic data associated to a quasi-ordinary projection
  11. The graded ring associated with finitely many monomial valuations
  12. Toric case
  13. Quasi-ordinary hypersurface case
  14. The Poincaré series as an integral with respect to the Euler characteristic
  15. Toric case
  16. ...and 6 more sections

Key Result

Proposition \oldthetheorem

(see Proposition 14, Pedro-resolucion-torica) The map $n: (Z_{\sigma, N}, o_\sigma) \rightarrow (S,0)$, corresponding to the ring extension is the normalization map.

Theorems & Definitions (44)

  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Definition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 34 more