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Effective regularity criteria for analytic mappings

Adamus J.

Demonstratio Mathematica

2010

Vol. 43, nr 2

235-262

We survey recent developments in the study of singularities of complex-analytic mappings from a local algebraic viewpoint. We present several effective criteria for various modes of regularity of complex-analytic mappings in terms of vertical components in fibred powers of the mappings.

Fibres of analytic maps

Kwieciński M., Tworzewski P.

Bulletin of the Polish Academy of Sciences. Mathematics

1999

Vol. 47, no 3

245-255

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.