On the structure of the fibers of truncation morphisms

TL;DR

This paper investigates the structure of fibers in truncation morphisms of arc and jet spaces over algebraic schemes, generalizing previous results and applying findings to determine bounds on motivic zeta function poles.

Contribution

It provides a comprehensive analysis of fiber structures in truncation morphisms, extending earlier work and enabling calculation of optimal pole bounds for motivic zeta functions.

Findings

Generalized previous results on fiber structures

Established bounds for poles of motivic zeta functions

Enhanced understanding of arc and jet space morphisms

Abstract

Let k be an algebraically closed field and let X be a separated scheme of finite type over k of pure dimension d. We study the structure of the fibres of the truncation morphisms from the arc space of X to jet spaces of X and also between jet spaces. Our results are generalizations of results of Denef, Loeser, Ein and Mustata. We will use them to find the optimal lower bound for the poles of the motivic zeta function associated to an arbitrary ideal.