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Arithmetic motivic
Poincaré series of toric varieties
Helena Cobo Pablos and Pedro Daniel González Pérez |
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Vol. 7 (2013), No. 2, 405–430
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Abstract |
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The arithmetic motivic Poincaré series of a variety 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 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 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
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Mathematical Subject Classification 2010
Primary: 14M25
Secondary: 14J17, 14B05
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Milestones
Received: 7 November 2011
Revised: 31 January 2012
Accepted: 3 March 2012
Published: 25 April 2013
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Authors |
| Helena Cobo Pablos | |
| Department of Mathematics University of Leuven Celestijnenlaan 200B B-3001 Heverlee Belgium |
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| 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 |
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