Vol. 89, No. 1, 1980

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On equisingular families of isolated singularities

Augusto Nobile

Vol. 89 (1980), No. 1, 151–161

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.

Mathematical Subject Classification 2000
Primary: 14H20
Received: 15 July 1977
Published: 1 July 1980
Augusto Nobile