## Segre-degenerate points form a semianalytic set

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## 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.## References

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## Additional Information

**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
- © Copyright 2022 by the author under Creative Commons Attribution 3.0 License (CC BY 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B
**9**(2022), 159-173 - MSC (2020): Primary 32C07; Secondary 32B20, 14P15
- DOI: https://doi.org/10.1090/bproc/99
- MathSciNet review: 4407043