Vol. 7, No. 2, 2013

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Arithmetic motivic Poincaré series of toric varieties

Helena Cobo Pablos and Pedro Daniel González Pérez

Vol. 7 (2013), No. 2, 405–430

The arithmetic motivic Poincaré series of a variety V defined over a field of characteristic zero is an invariant of singularities that was introduced by Denef and Loeser by analogy with the Serre–Oesterlé series in arithmetic geometry. They proved that this motivic series has a rational form that specializes to the Serre–Oesterlé series when V is defined over the integers. This invariant, which is known explicitly for a few classes of singularities, remains quite mysterious. In this paper, we study this motivic series when V is an affine toric variety. We obtain a formula for the rational form of this series in terms of the Newton polyhedra of the ideals of sums of combinations associated to the minimal system of generators of the semigroup of the toric variety. In particular, we explicitly deduce a finite set of candidate poles for this invariant.

arithmetic motivic Poincaré series, toric geometry, singularities, arc spaces
Mathematical Subject Classification 2010
Primary: 14M25
Secondary: 14J17, 14B05
Received: 7 November 2011
Revised: 31 January 2012
Accepted: 3 March 2012
Published: 25 April 2013
Helena Cobo Pablos
Department of Mathematics
University of Leuven
Celestijnenlaan 200B
B-3001 Heverlee
Pedro Daniel González Pérez
Instituto de Ciencias Matemáticas
Departamento de Álgebra
Facultad de Ciencias Matemáticas
Universidad Complutense de Madrid
Plaza de las Ciencias 3
28040 Madrid