Quasi Ordinary Singularities, Essential Divisors and Poincare Series

TL;DR

This paper introduces a new Poincaré series invariant for quasi-ordinary hypersurface singularities, showing it is rational and encodes topological and analytical information, including characteristic monomials.

Contribution

It defines a Poincaré series for quasi-ordinary singularities using monomial valuations, proving its rationality and its role as a complete invariant for the embedded topological type.

Findings

Poincaré series is a rational function with integer coefficients.

The series is an analytic invariant of the singularity.

In the quasi-ordinary case, it determines the characteristic monomials.

Abstract

We define Poincar\'e 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 Poincar\'e 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 Poincar\'e 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 Poincar\'e 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.