The monodromy conjecture for ideals

Central to this thesis is a proof of the monodromy conjecture for a specific kind of binomial ideals in arbitrary dimension that define the so-called space monomial curves with a plane semigroup. These curves arise in a natural way as the special fibers of certain deformations of plane curve singularities. Roughly speaking, the monodromy conjecture for a general ideal states that the poles of the motivic, or some related, Igusa zeta function induce eigenvalues of monodromy. To date, this conjecture has only been proven for ideals in two variables, and it is one of the most intriguing open problems in singularity theory. 

Our proof starts with a computation of the motivic Igusa zeta function: by studying the jet schemes of a space monomial curve, we obtain a closed formula for the motivic Igusa zeta function as well as a complete list of its poles. Next, we simplify the problem of computing the monodromy eigenvalues by considering a space monomial curve as a Cartier divisor on a generic embedding surface. In this way, we can use an A'Campo formula in terms of an embedded Q-resolution of the latter pair to determine the monodromy eigenvalues. Finally, we combine all results to conclude the monodromy conjecture for a space monomial curve. 

Since the above-mentioned simplification to find the monodromy eigenvalues is valid for curves defined by a larger class of ideals, we prove it in a more general setting; this makes it possibly useful for other instances of the monodromy conjecture. In addition, we show an interesting relation between the motivic Igusa zeta function of these more general curves in the affine space, on the one hand, and on a generic surface, on the other hand. This yields a close connection between the monodromy conjectures in both settings. In particular, we establish the monodromy conjecture for a space monomial curve on a generic surface. 

As a side project, we take a look at the link of a generic embedding surface of a space monomial curve. More precisely, by using a good Q-resolution of a generic surface singularity, we determine necessary and sufficient conditions for its link to be a rational or integral homology sphere.