A remark on symbolic powers
Volume 249 / 2019
Studia Mathematica 249 (2019), 111-116
MSC: Primary 13A10; Secondary 32A10.
DOI: 10.4064/sm180502-12-6
Published online: 8 April 2019
Abstract
The symbolic powers ${\mathcal I}^{(p)}$ of an ideal ${\mathcal I}\subset {\mathcal {O}}(X)$ generated by holomorphic functions $g_1,\ldots ,g_r\in {\mathcal {O}}(Y)$ on a Stein manifold $X\Subset Y$ are shown to satisfy ${\mathcal I}^{(p+q)}\subset {\mathcal I}^p$ for $q=\min\{\dim X,r-1\}$ and all $p\in {\mathbb N}$, which contributes to the so-called containment problem. In particular, this is true if ${\mathcal I}$ is the vanishing ideal of a finite set.
