Poincaré series of a toric variety

Abstract

For an affine toric variety X we compute the Poincaré series of the multi-index filtration defined by a finite number of monomial divisorial valuations on the ring O X , 0 . We give an alternative description of the Poincaré series as an integral with respect to the Euler characteristic over the projectivization of the space of germs O X , 0 . In particular we study divisorial valuations on the ring O C d , 0 that arise by considering toric constellations. We give an explicit formula for the Poincaré series and a nice geometric description. This generalizes an expression of the Poincaré series for curves and rational surface singularities.