# The motivic Igusa zeta function of a space monomial curve with a plane semigroup

## TL;DR

This paper computes the motivic Igusa zeta function for a space monomial curve arising from a family of plane branches, revealing its poles and their relation to the generic fiber.

## Contribution

It provides a closed-form expression for the motivic zeta function of a space monomial curve and analyzes its poles, linking them to the generic fiber in the family.

## Key findings

- Derived the irreducible components of jet schemes for the curve.
- Obtained a closed formula for the motivic Igusa zeta function.
- Showed the number of poles matches that of the generic fiber.

## Abstract

In this article, we compute the motivic Igusa zeta function of a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a complex plane branch. To this end, we determine the irreducible components of the jet schemes of such a space monomial curve. This approach does not only yield a closed formula for the motivic zeta function, but also allows to determine its poles. We show that, while the family of the jet schemes of the fibers is not flat, the number of poles of the motivic zeta function associated with the space monomial curve is equal to the number of poles of the motivic zeta function associated with a generic curve in the family.