  Let $X^n \subset P^N$ be a nonsingular, nondegenerate projective variety of
dimension $n$ and codimension $N-n \ge 2$. Let $|C_X|$ be the linear system
determined by the double-point divisor obtained by generically projecting $X$
to a hypersurface in $P^{n+1}$. We classify those varieties for which $C_X$ is
not ample, or equivalently, does not separate points of $X$. We call such
varieties Roth varieties and prove that they exist for all dimensions $n \ge 2$
and give a description of their properties. For example, in many cases Roth
varieties are Castelnuovo varieties. Positivity results for the double-point
divisor are analogous to positivity results for the ramification divisor which
are studied in adjunction theory.
