We use the difference of two analytic invariants of a curve singularity - the multiplicity of its Jacobian ideal and its complete intersection discrepancy, to obtain a fiberwise multiplicity control of equisingularity for families of reduced curves over smooth parameter spaces of arbitrary positive dimension. Our result generalizes a result of Briançon, Galligo and Granger for one-parameter families of curves. If time permits I will discuss the Zariski upper semicontinuity of the Milnor number.
This is a report on a joint work with Bengus-Lasnier and Gaffney. | 
