Segre-Degenerate Points Form a Semianalytic Set
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909208259067904 |
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| author | Lebl, Jiri |
| author_facet | Lebl, Jiri |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2102_07025 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Segre-Degenerate Points Form a Semianalytic Set Lebl, Jiri Complex Variables Algebraic Geometry 32C07 (Primary), 32B20, 14P15 (Secondary) 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. |
| title | Segre-Degenerate Points Form a Semianalytic Set |
| topic | Complex Variables Algebraic Geometry 32C07 (Primary), 32B20, 14P15 (Secondary) |
| url | https://arxiv.org/abs/2102.07025 |