# Proceedings of the American Mathematical Society Series B

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society Series B (BPROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 2330-1511

The 2020 MCQ for Proceedings of the American Mathematical Society Series B is 0.84.

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## Segre-degenerate points form a semianalytic setHTML articles powered by AMS MathViewer

by Jiří Lebl
Proc. Amer. Math. Soc. Ser. B 9 (2022), 159-173

## 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 need not be complex; \$X_{[n]}\$ can, for instance, be a real line.
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• Jiří Lebl
• Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
• MR Author ID: 813143
• ORCID: 0000-0002-9320-0823
• Email: lebl@okstate.edu
• Received by editor(s): April 2, 2021
• Received by editor(s) in revised form: August 2, 2021, and August 12, 2021
• Published electronically: April 12, 2022
• Additional Notes: The author was supported in part by Simons Foundation collaboration grant 710294.
• Communicated by: Harold P. Boas