  We prove a stratification result for certain families of $n$-dimensional
(complete algebraic) multiplicative character sums. The character sums we
consider are sums of products of $r$ multiplicative characters evaluated at
rational functions, and the families (with $nr$ parameters) are obtained by
allowing each of the $r$ rational functions to be replaced by an "offset", i.e.
a translate, of itself. For very general such families, we show that the
stratum of the parameter space on which the character sum has maximum weight
$n+j$ has codimension at least $j\lfloor(r-1)/2(n-1)\rfloor$ for $1\le j\le
n-1$ and $\lceil nr/2\rceil$ for $j=n$.
