Arithmetic motivic Poincaré series of toric varieties

Cobo Pablos, Helena; González Pérez, Pedro Daniel

doi:10.2140/ant.2013.7.405

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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

DOI: 10.2140/ant.2013.7.405

Abstract

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.

Keywords

arithmetic motivic Poincaré series, toric geometry, singularities, arc spaces

Mathematical Subject Classification 2010

Primary: 14M25

Secondary: 14J17, 14B05

Milestones

Received: 7 November 2011

Revised: 31 January 2012

Accepted: 3 March 2012

Published: 25 April 2013

Authors

Helena Cobo Pablos
Department of Mathematics University of Leuven Celestijnenlaan 200B B-3001 Heverlee Belgium

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 Spain

Vol. 7, No. 2, 2013