Segre-degenerate points form a semianalytic set

We prove that the set of Segre-degenerate points a real-analytic subvariety X X in alttext="double-struck upper C Superscript n"> C n encoding="application/x-tex">{\mathbb {C}}^n is closed semianalytic set. It if coherent. More precisely, where germ Segre variety dimension alttext="k"> k encoding="application/x-tex">k or greater general, and for coherent , it . For hypersurface points, X Subscript left-bracket n right-bracket"> [ stretchy="false">] encoding="application/x-tex">X_{[n]} at most alttext="2 minus 4"> 2 − 4 encoding="application/x-tex">2n-4 If coherent, then complex (complex) alttext="n 2"> encoding="application/x-tex">n-2 Example hypersurfaces are given showing need not be also complex; can, instance, real line.