Department of Mathematics

The University of Western Ontario

Failure of some (important) properties of a holomorphic mapping manifests as degeneracies in the family of fibres of the mapping. Among these properties are openness and flatness. The first goal in my thesis is to develop criteria such that first, they effectively (i.e., computationally) detect such degeneracies in the family of fibres, and second, they are applicable to the case of mappings with singular targets. Particularly for 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 in the pullback of the mapping by the blowing-up. 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.