Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.10366 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866910048661274624 |
|---|---|
| author | Chardin, Marc D'Cruz, Clare |
| author_facet | Chardin, Marc D'Cruz, Clare |
| contents | 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 ${\mathbb P}^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_10366 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Curves in ${\mathbb P}^n$ of analytic spread at most $n$ Chardin, Marc D'Cruz, Clare Commutative Algebra 13A30, 14H50, 13D45 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 ${\mathbb P}^3$. |
| title | Curves in ${\mathbb P}^n$ of analytic spread at most $n$ |
| topic | Commutative Algebra 13A30, 14H50, 13D45 |
| url | https://arxiv.org/abs/2603.10366 |