${\mathbb P}^n$中分析维数不超过n的曲线
Curves in ${\mathbb P}^n$ of analytic spread at most $n$
AI总结：
研究${\mathbb P}^n$中分析维数不超过n的曲线，证明其极限深度为1且Rees环正则度至多1，纤维锥为Cohen-Macaulay。
摘要：
我们研究了${\mathbb P}^n$中维数为一的闭子方案X，这些方案在任意一点局部由至多n个方程定义，且$I_{\mathfrak{m}}$的分析维数不超过n，其中$I$是X的定义理想，${\mathfrak{m}}=(x_0, \ldots, x_n)$。在这种情况下，我们证明在温和条件下，所有$I_{\mathfrak{m}}$的幂都有正深度，因此$I_{\mathfrak{m}}$的极限深度为1，除非I是一个完全交集。此外，Rees环的正则度至多为一，纤维锥是Cohen-Macaulay的。这适用于定义于${\mathbb P}^3$中的每一个单项式曲线的理想。
英文摘要：
We study closed subschemes $X$ in ${\mathbb P}^n$ of dimension one, locally defined at any point by at most $n$ equations such that the analytic spread of $I_{\mathfrak{m}}$ is at most $n$, where $I \subseteq \Bbbk[x_0, \ldots, x_n] $ is the defining ideal of $X$ and ${\mathfrak{m}} = (x_0, \ldots, x_n)$. In this situation, we show that, under mild conditions, all the powers of $I_{\mathfrak{m}}$ have positive depth, hence the limit depth of $I_{\mathfrak{m}}$ is $1$ unless $I$ is a complete intersection. Moreover, the regularity of the Rees ring is at most one and the fiber cone is Cohen-Macaulay. This applies to every ideal defining a monomial curve in ${\mathbb P}^3$.
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