It is studied the foundations of
a theory of equisingularity for plane algebroid or complex analytic curves, reduced
everywhere except at one point. A definition of equivalence, or equisingularity, for
two such curves, involving resolution by means of quadratic transformations is given
as well as several definitions of the concept of equisingular one-parameter family of
such singularities are proposed. The theory is related to both the theory of
equisingularity for plane, reduced curves and that for ideals with finite support. The
different notions of equisingularity for families are compared and a number of
examples are presented.
