Motivic zeta function of the Hilbert schemes of points on a surface

Abstract

Let K be a discretely-valued field. Let X → Spec K be a surface with trivial canonical bundle. In this paper we construct a weak Néron model of the schemes Hilb n (X) over the ring of integers R ⊆ K. We exploit this construction in order to compute the Motivic Zeta Function of Hilb n (X) in terms of Z X . We determine the poles of Z Hilb n (X) and study its monodromy property, showing that if the monodromy conjecture holds for X then it holds for Hilb n (X) too. ExcerptumSit K corpus cum absoluto ualore discreto. Sit X → Spec K leuigata superficies cum canonico fasce congruenti O X . In hoc scripto defecta Neroniensia paradigmata Hilb n (X) schematum super annulo integrorum in K corpo, R ⊂ K, constituimus. Ex hoc, Functionem Zetam Motiuicam Z Hilb n (X) , dato Z X , computamus. Suos polos statuimus et suam monodromicam proprietatem studemus, coniectura monodromica, quae super X ualet, ualere super Hilb n (X) quoque demostrando.This project has received funding from the European Union Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie grant agreement No 801199. Grothendick ring of varietiesIn this section we introduce the rings containing the coefficients of the formal series we shall study later on.