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Journal of the London Mathematical Society Advance Access originally published online on April 30, 2009
Journal of the London Mathematical Society 2009 79(3):780-802; doi:10.1112/jlms/jdp014
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© 2009 London Mathematical Society

Quasi-ordinary singularities, essential divisors and Poincaré series

P. D. González Pérez

Instituto de Ciencias Matemáticas-CSIC-UAM-UC3M-UCM
Departamento de Algebra
Facultad de Ciencias Matemáticas
Universidad Complutense de Madrid
Plaza de las Ciencias 3
28040 Madrid
Spain
pgonzalez@mat.ucm.es

F. Hernando

School of Mathematical Sciences
Aras Na Laoi
University of Cork
Ireland

We define Poincaré series associated to a germ (S, 0) of toric or analytically irreducible quasi-ordinary hypersurface singularity, by a finite sequence of monomial valuations such that at least one of them is centered at the point 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 also be defined as an integral with respect to the Euler characteristic of a function defined by the valuations, over the projectivization of the analytic algebra of the singularity. In particular, the Poincaré series associated to the set of divisorial valuations of 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 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.

2000 Mathematics Subject Classification 14J17 (primary), 32S10, 14M25 (secondary).

González Pérez is supported by Programa Ramón y Cajal and MTM2007-6798C0202 grant of Ministerio de Educación y Ciencia (MEC), Spain. Hernando is supported by the Claude Shannon Institute, Science Foundation of Ireland Grant 06/MI/006, by grant MTM2007-64704 of MEC, (Spain) and grant VA025/07 of Junta de Castilla y León, Spain.

Received August 1, 2008; revised February 18, 2009; published online April 30, 2009.


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