The symbolic and divisorial analytic spreads are finite

Abstract

Let $R$ be a ring that is essentially of finite type over a field and $I\subseteq R$ be an ideal. In this article it is shown that the minimal number of generators of the symbolic powers $I^{(n)}$ is bounded above by a polynomial in $n$. Furthermore, if $R$ is assumed to be a domain and $\mathcal I=\{I_n\}_{n\ge 0}$ is an $\mathbb{R}$-divisorial filtration, then the minimal number of generators of the ideals $I_n$ is again bounded above by a polynomial in $n$.