Abstract
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.
Highlights
We gathered here a collection of criteria for various modes of regularity of analytic mappings, like Gabrielov regularity, openness, or flatness
We say that Wg is geometric vertical if any sufficiently small representative of W,£ is mapped into a nowhere dense subset of a neighbourhood of 77 in Y. (The notion of vertical component was introduced by Kwiecinski [17], to mean what we call a geometric vertical component.)
As we show in Example 4.2, there are examples of bad behaviour of analytic mappings that can be detected by means of geometric vertical components but not by the algebraic vertical ones
Summary
Recall that a morphism Yv of germs of analytic spaces is called flat when the pull-back homomorphism : OY,v —> Ox,£ makes the local ring Ox,£ into a flat Oyi7?-module. (Otherwise, for instance, each of the irreducible components of the set X — {xy = 0} would be an algebraic vertical component for the identity mapping idx : X —> X, which is absurd.) This is in contrast with Theorem 1.8, where any weakening of the regularity assumption on the target space leads to an open problem. As of today, it is not even known whether, for a singular irreducible Yv, there exists a constant N (depending only on Yv) such that non-flatness of a morphism ip^ : —> Yv could be detected by vertical components in the iV-fold fibred power of ip^. The last section contains computable criteria for openness and flatness of polynomial mappings, derived from Theorems 1.10 and 7.1
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