Motivic generating series for toric surface singularities

Details Motivic generating series for toric surface singularities Motivic generating series for toric surface singularities

2004-02-12

arxiv.org/abs/math/0402196v1

Mathematics

Algebraic Geometry

18 pages

Scientific paper

10.1017/S0305004105008418

Lejeune-Jalabert and Reguera computed the geometric Poincare series P_{geom}(T) for toric surface singularities. They raise the question whether this series equals the arithmetic Poincare 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 Poincare 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 multiplicity of the singularity. In some sense, this is the only difference between Q_{geom}(T) and P_{geom}(T).

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