Curves in ${\mathbb{P}}^n$ of analytic spread at most $n$

Abstract

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 $I\subseteq \mathbb{k}[x_0,\dots ,x_n]$ is the defining ideal of $X$ and $\mathfrak{m}=(x_0,\dots ,x_n)$. 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 ${\mathbb{P}}^3$.