Berkovich skeleta and birational geometry

We give a survey of joint work with Mircea Musta\c{t}\u{a} and Chenyang Xu on the connections between the geometry of Berkovich spaces over the field of Laurent series and the birational geometry of one-parameter degenerations of smooth projective varieties. The central objects in our theory are the weight function and the essential skeleton of the degeneration. We tried to keep the text self-contained, so that it can serve as an introduction to Berkovich geometry for birational geometers.Comment: These are expanded lecture notes of a talk at the Simons Symposium on Non-Archimedean Geometry and Tropical Geometry (March 31-April 6, 2013). They have been submitted to the conference proceeding

Motivic generating series for toric surface singularities

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).Comment: 18 page