Segre-degenerate points form a semianalytic set

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.