On some curve singularity invariants and reductions

TL;DR

This paper investigates the behavior of curve singularity invariants under reduction modulo a maximal ideal and introduces a global zeta function via motivic integration that generalizes known singularity zeta functions.

Contribution

It defines a new global zeta function for curve singularities using motivic integration, linking it to existing local zeta functions through reduction.

Findings

Behavior of the value semigroup under reduction

Definition of a global zeta function via motivic integration

Reduction of the global zeta function to known local zeta functions

Abstract

This paper deals with the study of the behaviour of the value semigroup of a curve singularity define over a global field reduced modulo a maximal ideal. We also define a global zeta function of the curve by means of motivic integration over a suitable ring of ad\'eles, whose reduction modulo a maximal ideal will coincide with some already known zeta functions of the singularity.