Segre-Degenerate Points Form a Semianalytic Set

Abstract

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.

Theorems & Definitions (40)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• Definition 2.2: see e.g. Loja:semiBM:semisub
• Example 2.3
• Lemma 2.4
• proof
• Example 2.5
• Proposition 3.1
• proof