In this paper we study
L(S)-tuples of matrices, a class of k-tuples that includes as special cases the L-pairs
studied by Motzkin and Taussky, and the l-pairs defined by Taussky for
complex elements and diagonable first matrix. Some light on these concepts
and a few relevant results are produced by linking them to properties of
algebraic hypersurfaces and especially curves with respect to an exterior
point.