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Basic properties of a definition
of equisingularity for families of (algebraic, analytic or algebroid) varieties, singular
along a given section, are studied. The equisingularity condition is: given a
family p : X → Y , with a section s, it is required that the natural morphism
E → Y be flat, where E is the exceptional divisor of the blowing-up of X
with center the product of the ideal defining s and the relative Jacobian
ideal.
The following results hold: (a) This condition is invariant under base
change (b) It implies equimultiplicity, the validity of the Whitney conditions
and topological triviality along s (c) If Y is reduced, the condition holds
over a dense open set of Y , whose complement is a subvariety of Y (d) If Y
is smooth and the fibers of p are plane curves, this definition agrees with
Zariski’s.
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