Abstract
Roughly speaking, the monodromy conjecture for a singularity states that every pole of its motivic Igusa zeta function induces an eigenvalue of its monodromy. In this note, we determine both the motivic Igusa zeta function and the eigenvalues of monodromy for a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a plane branch. In particular, this yields a proof of the monodromy conjecture for such a curve.
Highlights
The monodromy conjecture for ideals predicts a relation between two invariants associated with an ideal, one originating from number theory and the other from differential topology
The monodromy conjecture for a singularity states that every pole of its motivic Igusa zeta function induces an eigenvalue of its monodromy
We determine both the motivic Igusa zeta function and the eigenvalues of monodromy for a space monomial curve that appears as the special fiber of an equisingular family whose generic fiber is a plane branch
Summary
The monodromy conjecture for ideals predicts a relation between two invariants associated with an ideal, one originating from number theory and the other from differential topology. These curves arise as the special fibers of equisingular families of curves whose generic fibers are isomorphic to some plane branch After introducing these curves in more detail, we show the results, without proofs, obtained in [10]; by studying the jet schemes of a space monomial curve Y ⊂ Cg+1, we are able to compute the motivic Igusa zeta function and to determine its poles, see Theorem 3 and Theorem 4, respectively. We explain the approach of [6], again without proofs, to reduce the problem of studying the monodromy eigenvalues associated with Y ⊂ Cg+1 by considering Y as a Cartier divisor on a generic embedding surface S This way, we can use an A’Campo formula in terms of an embedded Q-resolution of Y ⊂ S to compute the monodromy zeta function of Y , see Theorem 8. We combine all results to conclude the monodromy conjecture for a space monomial curve Y ⊂ Cg+1 in Theorem 10
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
