  We consider families of parameterizations of reduced curve singularities over
a Noetherian base scheme and prove that the delta invariant is semicontinuous.
In our setting, each curve singularity in the family is the image of a
parameterization and not the fiber of a morphism. The problem came up in
connection with the right-left classification of parameterizations of curve
singularities defined over a field of positive characteristic. We prove a bound
for right-left determinacy of a parameterization in terms of delta and the
semicontinuity theorem provides a simultaneous bound for the determinacy in a
family. The fact that the base space can be an arbitrary Noetherian scheme
causes some difficulties but is (not only) of interest for computational
purposes.
