Quasi‐ordinary singularities, essential divisors and Poincaré series

Abstract

Abstract. We define Poincaré series associated to a toric or analytically irreducible quasiordinary 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 multigraded 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 corresponding 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.

Finite families of plane valuations: value semigroup, graded algebra and Poincaré series

“…An explicit computation for the plane divisorial case can be found in [25]; as we shall see, the Poincaré series is very close to that attached to the semigroup of the valuation or to the Poincaré series of the analytically irreducible germ of curve provided by a general element of the valuation [30]. One can found many papers studying Poincaré series for singularities (which need not to correspond to the irreducible case), some of them are [8,9,15,28,40,45].…”

Geometric motivic Poincaré series of quasi-ordinary singularities

“…We have that P (S,0) geom (T ) = (1 − T ) −1 + P(S θ 1 ) + P(S θ 2 ) + P(S). The motivic volume is 18 (1 − L 4 )(1 − L 20 ) .…”

“…singularities is also of interest to test and study various open questions and conjectures for singularities in general, particularly in the hypersurface case (see [32]). In many cases the results passed by using the fractional power series parametrizations of these singularities, which allow explicit computations combining analytic, topological and combinatorial arguments (see for instance [29,13,37,33,2,20]). It is natural to investigate new invariants of singularities, such as those arising in the development of motivic integration, on this class of singularities with the hope to extend the methods or results to wider classes (for instance by passing through Jung's approach).…”