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

TL;DR

This paper investigates one-dimensional subschemes in projective space with bounded analytic spread, showing that under mild conditions, their defining ideals have favorable algebraic properties, including positive depth of powers and Cohen-Macaulay fiber cones.

Contribution

It establishes new properties of ideals defining curves in projective space with analytic spread at most n, including depth stability and Cohen-Macaulayness of associated graded structures.

Findings

Powers of the ideal have positive depth under mild conditions.

The limit depth of the ideal is 1 unless it is a complete intersection.

The Rees algebra has regularity at most one and the fiber cone is Cohen-Macaulay.

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 \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 ${\mathbb P}^3$.