The symbolic defect of an ideal

Abstract

Let $I$ be a homogeneous ideal of $\mathbb{k}[x_0,\dots ,x_n]$. To compare $I^{(m)}$, the $m$-th symbolic power of $I$, with $I^m$, the regular $m$-th power, we introduce the $m$-th symbolic defect of $I$, denoted $\mathrm{sdefect}(I,m)$. Precisely, $\mathrm{sdefect}(I,m)$ is the minimal number of generators of the $R$-module $I^{(m)}/I^m$, or equivalently, the minimal number of generators one must add to $I^m$ to make $I^{(m)}$. In this paper, we take the first step towards understanding the symbolic defect by considering the case that $I$ is either the defining ideal of a star configuration or the ideal associated to a finite set of points in ${\mathbb{P}}^2$. We are specifically interested in identifying ideals $I$ with $\mathrm{sdefect}(I,2)=1$.

To appear in Journal of Pure and Applied Algebra; revised at referees' suggestion. Fixed typos and clarified writing, included additional references, shortened proof of Thm 6.3