  We consider families of schemes over arbitrary fields resp. analytic
varieties with finitely many (not necessarily reduced) isolated non-normal
singularities, in particular families of generically reduced curves. We define
a modified delta invariant for isolated non-normal singularities of any
dimension that takes care of embedded points and prove that it behaves upper
semicontinuous in flat families parametrized by an arbitrary principal ideal
domain. Moreover, if the fibers contain no isolated points, then the familly
admits a fiberwise normalization iff the delta invariant is locally constant.
The results generalize results by Teissier and Chiang-Hsieh--Lipman for
families of reduced curve singularities and provide possible improvements for
algorithms to compute the genus of a curve.
