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

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.

The monodromy conjecture for a space monomial curve with a plane semigroup

“…In [MVV,Example 4.1], it was shown that the motivic Igusa zeta function of Y 1 has three poles: L 2 , L are poles of Z mon Y 1 ,p 1 (t).…”

“…The poles of the motivic zeta function of Y 2 were also computed in [MVV,Example 4.1]: they are given by L 3 , L . Similarly as in the previous example, it is easy to see that they all induce eigenvalues of monodromy associated with Y 2 .…”