Electronic Thesis and Dissertation Repository

Degree

Doctor of Philosophy

Program

Mathematics

Supervisor(s)

Janusz Adamus

Abstract

Failure of some (important) properties of a holomorphic mapping manifests itself as degeneracies in the family of fibres of the mapping. Among these properties are openness and flatness. The first goal in this thesis is to develop criteria that allow one to effectively (i.e., computationally) detect such degeneracies in the family of fibres, and in addition, that are applicable to the case of mappings with singular targets. Particularly regarding flatness, no such algorithms that work in the general setting of singular targets were known before. We prove that a mapping (with a locally irreducible target) is flat (resp. open) if and only if no (resp. isolated) irreducible component is mapped to the origin after pulling back the mapping by the blowing-up. We give also a generalization of previous flatness criteria of Auslander's type to the case of singular bases. The second goal is to characterize different modes of such degeneracies. We take an index that measures the level of non-openness of mappings, and obtain some results on its behaviour.


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Mathematics Commons

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