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.

Normal surface singularities with an integral homology sphere link related to space monomial curves with a plane semigroup

“…In [16], the monodromy eigenvalues for such a space monomial curve Y ⊂ C g+1 with g ≥ 2 are investigated by considering Y as a Cartier divisor a generic embedding surface S ⊂ C g+1 . Together with the results from [17], this yields a proof of the monodromy conjecture for Y ⊂ C g+1 . In this article, we are interested in the topology of these generic embedding surface singularities (S, 0) ⊂ (C g+1 , 0).…”

“…The monodromy conjecture for these space monomial curves Y ⊂ C g+1 with g ≥ 2 is proven in [16] together with [17]; an overview of these two articles can be found in the short note [14]. Roughly speaking, the monodromy conjecture for Y ⊂ C g+1 states that the poles of the motivic, or related, Igusa zeta function of Y induce monodromy eigenvalues of Y .…”

“…Roughly speaking, the monodromy conjecture for Y ⊂ C g+1 states that the poles of the motivic, or related, Igusa zeta function of Y induce monodromy eigenvalues of Y . The computation of the motivic Igusa zeta function and its poles is the main subject of [17]; the study of the monodromy eigenvalues and proof of the monodromy conjecture can be found in [16]. A key ingredient in this proof is the consideration of the curve Y as the Cartier divisor { f i = 0} for some i ∈ {1, .…”

Resolving singularities of curves with one toric morphism

“…In addition, a resolution obtained by a torific embedding is often easier to use than classsical Hironaka-type resolutions for computing subtle invariants of singularities like motivic or topological zeta functions, monodromy zeta functions, log canonical thresholds and jumping numbers of multiplier ideals (see [19,20,35]). The resolutions of singularities obtained by the classical approach are often complex to handle and it is a difficult problem to link invariants of resolution of singularities like Hironaka's order of ideals to subtle invariants of singularities such as those that we have just mentioned.…”