On delta for parameterized curve singularities

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