  In this work we study equisingularity in a one-parameter flat family of
generically reduced curves. We consider some equisingular criteria as
topological triviality, Whitney equisingularity and strong simultaneous
resolution. In this context, we prove that Whitney equisingularity is
equivalent to strong simultaneous resolution and it is also equivalent to the
constancy of the Milnor number and the multiplicity of the fibers. These
results are extensions to the case of flat deformations of generically reduced
curves, of known results on reduced curves. When the family $(X,0)$ is
topologically trivial, we also characterize Whitney equisingularity through
Cohen-Macaulay property of a certain local ring associated to the parameter
space of the family.
