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

TL;DR

This paper constructs a weak Néron model for Hilbert schemes of points on a surface with trivial canonical bundle over a discretely-valued field, computes their motivic zeta functions, and explores the monodromy conjecture's implications.

Contribution

It introduces a method to compute the motivic zeta function of Hilbert schemes of points on such surfaces and relates their monodromy properties to those of the original surface.

Findings

Constructed weak Néron models for Hilbert schemes

Expressed motivic zeta functions in terms of the base surface

Established the link between monodromy conjecture for the surface and its Hilbert schemes

Abstract

Let $K$ be a discretely-valued field. Let $X\rightarrow Spec K$ be a surface with trivial canonical bundle. In this paper we construct a weak N\'eron model of the schemes $Hilb^n(X)$ over the ring of integers $R\subseteq K$. 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. Sit $K$ corpus cum absoluto ualore discreto. Sit $ X\rightarrow Spec K$ leuigata superficies cum canonico fasce congruenti $\mathcal{O}_X$. In hoc scripto defecta Neroniensia paradigmata $Hilb^n(X)$ schematum super annulo integrorum in $K$ corpo, $R \subset K$, 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.