  We prove that the set of Segre-degenerate points of a real-analytic
subvariety $X$ in ${\mathbb{C}}^n$ is a closed semianalytic set. It is a
subvariety if $X$ is coherent. More precisely, the set of points where the germ
of the Segre variety is of dimension $k$ or greater is a closed semianalytic
set in general, and for a coherent $X$, it is a real-analytic subvariety of
$X$. For a hypersurface $X$ in ${\mathbb{C}}^n$, the set of Segre-degenerate
points, $X_{[n]}$, is a semianalytic set of dimension at most $2n-4$. If $X$ is
coherent, then $X_{[n]}$ is a complex subvariety of (complex) dimension $n-2$.
Example hypersurfaces are given showing that $X_{[n]}$ need not be a subvariety
and that it also needs not be complex; $X_{[n]}$ can, for instance, be a real
line.
