  The $X$-rank of a point $p$ in projective space is the minimal number of
points of an algebraic variety $X$ whose linear span contains $p$. This notion
is naturally submultiplicative under tensor product. We study geometric
conditions that guarantee strict submultiplicativity. We prove that in the case
of points of rank two, strict submultiplicativity is entirely characterized in
terms of the trisecant lines to the variety. Moreover, we focus on the case of
curves: we prove that for curves embedded in an even-dimensional projective
space, there are always points for which strict submultiplicativity occurs,
with the only exception of rational normal curves.
