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
$\operatorname{sdefect}(I,2) = 1$.
