Fibres of Analytic Maps

摘要

We prove that if a generically finite algebraic map has an isolated special fibre of small positive dimension then it has many points in the generic fibre. We also generalize this result to the case of analytic maps and non-isolated special fibres. The proofs of these theorems are based on a systematic study of the geometry of fibred powers. The presence of "vertical" irreducible components in fibred powers provides topological information. For example, their absence is equivalent to the map being open.