Segre-degenerate points form a semianalytic set

Abstract

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.