  We find conditions that guarantee that a decomposition of a generic
third-order tensor in a minimal number of rank-$1$ tensors (canonical polyadic
decomposition (CPD)) is unique up to permutation of rank-$1$ tensors. Then we
consider the case when the tensor and all its rank-$1$ terms have symmetric
frontal slices (INDSCAL). Our results complement the existing bounds for
generic uniqueness of the CPD and relax the existing bounds for INDSCAL. The
derivation makes use of algebraic geometry. We stress the power of the
underlying concepts for proving generic properties in mathematical engineering.
