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