Segre退化点构成一个半解析集
Segre-Degenerate Points Form a Semianalytic Set
AI总结：
该研究证明复n维空间中实解析子簇的Segre退化点集为闭半解析集，明确了凝聚子簇、超曲面情形下该集合的性质与维数，并通过示例说明其不一定为子簇或复集。
摘要：
我们证明了复n维空间${\mathbb{C}}^n$中实解析子簇$X$的Segre退化点集是闭半解析集；若$X$是凝聚的，则该集合是子簇。更确切地说，一般情况下，Segre簇的芽维数大于等于$k$的点集是闭半解析集，而对于凝聚的$X$，它是$X$的实解析子簇。对于${\mathbb{C}}^n$中的超曲面$X$，Segre退化点集$X_{[n]}$是维数至多为$2n-4$的半解析集；若$X$是凝聚的，则$X_{[n]}$是复维数为$n-2$的复子簇。文中给出的超曲面示例表明，$X_{[n]}$不一定是子簇，也不一定是复的，例如它可以是一条实直线。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
