Motivic Poincar\'e series, toric singularities and logarithmic jacobian ideals

TL;DR

This paper provides an explicit description of the motivic Poincaré series for affine toric varieties, linking it to Newton polyhedra of logarithmic jacobian ideals, and identifies potential poles in the series.

Contribution

It offers a new explicit formula for the motivic Poincaré series of affine toric varieties using logarithmic jacobian ideals and Newton polyhedra.

Findings

Explicit formula for motivic Poincaré series of affine toric varieties

Identification of candidate poles in the series

Connection between series and Newton polyhedra of logarithmic jacobian ideals

Abstract

The geometric motivic Poincar\'e series of a variety, which was introduced by Denef and Loeser, takes into account the classes in the Grothendieck ring of the sequence of jets of arcs in the variety. Denef and Loeser proved that this series has a rational form. We describe it in the case of an affine toric variety of arbitrary dimension. The result, which provides an explicit set of candidate poles, is expressed in terms of the sequence of Newton polyhedra of certain monomial ideals, which we call logarithmic jacobian ideals, associated to the modules of differential forms with logarithmic poles outside the torus of the toric variety.