Poincaré series of a rational surface singularity

Alexander polynomials and Poincaré series of sets of ideals

“…. , E r } of components of the exceptional divisor D. Again, if S = C 2 or if all curves L described above are Cartier divisors on (S, 0), this filtration coincides with the divisorial filtration studied in [9] and [4]. Otherwise this is not the case.…”

“…In a similar way one can prove versions of the main statements from [4] and [7] for ideals in the ring of functions on a rational surface singularity or on their universal abelian covers. Let (S, 0) be a rational surface singularity and let π : (X, D) → (S, 0) be its resolution.…”

“…those for which σ m δσ v σ is an integer for any δ ∈ Γ) with the change of variables t d i → t i is the Poincaré series of the filtration on Ø S,0 corresponding to the set of ideals {I i } (cf. [4], [5]). Remark.…”

A filtration defined by arcs on a variety

“…A convenient way to see this is to express both Poincaré series as certain integrals with respect to the Euler characteristic over the projectivization PO V,0 of the space O V,0 of germs of functions on (V, 0). The definition of such an integral can be found, e.g., in [CDG2,CDG3]. It is inspired by the notion of motivic integration (see, e.g., [DL]) and in some sense dual to it.…”

“…The proof in [CDG3] is formulated for surface singularities, but it can be easily extended to an arbitrary collection of finitely determined valuations…”