Segre-Degenerate Points Form a Semianalytic Set

TL;DR

This paper proves that the set of Segre-degenerate points in real-analytic subvarieties of complex space is a closed semianalytic set, with specific properties for hypersurfaces and coherence conditions, including examples illustrating its complexity.

Contribution

It establishes the semianalytic nature of Segre-degenerate points and characterizes their structure in various cases, including hypersurfaces and coherent varieties.

Findings

Segre-degenerate points form a closed semianalytic set.

For hypersurfaces, the set has dimension at most 2n-4.

In coherent cases, the set is a complex subvariety of dimension n-2.

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.