Motivic generating series for toric surface singularities

Abstract

Lejeune–Jalabert and Reguera computed the geometric Poincaré series $P_{geom}(T)$ for toric surface singularities. They raised the question whether this series equals the arithmetic Poincaré series. We prove this equality for a class of toric varieties including the surfaces, and construct a counterexample in the general case. We also compute the motivic Igusa Poincaré series $Q_{geom}(T)$ for toric surface singularities, using the change of variables formula for motivic integrals, thus answering a second question of Lejeune–Jalabert and Reguera's. The series $Q_{geom}(T)$ contains more information than the geometric series, since it determines the dimension of the tangent space at the singularity. In some sense, this is the only difference between $Q_{geom}(T)$ and $P_{geom}(T)$.