Jumping numbers in higher dimensions: computation and contribution by exceptional divisors
Spronggetallen in hogere dimensies: berekening en bijdrage door exceptionele divisoren
Baumers, Hans; S0193510
In this thesis, we investigate how certain results for jumping numbers on surfaces can be generalized to higher dimensions. We focus on two particular problems. Firstly, we generalize the algorithm by Alberich-Carramiñana, Àlvarez Montaner and Dachs Cadefau for jumping numbers of ideals on surfaces with rational singularities. We introduce antieffective divisors in order to develop an algorithm that generates a small set of numbers, called supercandidates, containing the jumping numbers. Every supercandidate comes with a divisor, the minimal jumping divisor. This divisor gives an upper bound for possible divisors critically contributing the supercandidate as a jumping number. If the minimal jumping divisor is irreducible, which is often the case, the supercandidate is a jumping number. We also provide some other techniques, so that this can be easily checked in many other cases. Secondly, we try to generalize the results of Smith and Thompson and Tucker on contribution of jumping numbers by exceptional divisors. In particular, we study the relation between contributing divisors, the log canonical model and the intersection configuration in a log resolution. We show that if an exceptional divisor belongs to certain isomorphism classes, it contributes jumping numbers if and only if it survives in the log canonical model, and that we can see from the intersection configuration whether this is the case. We provide some counterexamples to show that this result does not hold in general. In particular, we construct an example of an exceptional divisor that does not contribute jumping numbers, but survives in the log canonical model.