# The symbolic defect of an ideal Academic Article

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### 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\$.

• June 2019