The symbolic defect of an ideal

TL;DR

This paper introduces the concept of symbolic defect to measure the difference between symbolic and regular powers of an ideal, focusing on specific geometric configurations and identifying cases where this defect equals one.

Contribution

It defines the symbolic defect and investigates its behavior for ideals of star configurations and finite point sets in projective space, especially when the defect is one.

Findings

Defined the m-th symbolic defect of an ideal.

Analyzed the case when the defect is one for specific geometric ideals.

Initiated the study of symbolic defect in algebraic geometry contexts.

Abstract

Let $I$ be a homogeneous ideal of $\Bbbk[x_0,\ldots,x_n]$. 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 $\operatorname{sdefect}(I,m)$. Precisely, $\operatorname{sdefect}(I,m)$ 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 $\operatorname{sdefect}(I,2) = 1$.