Let \(I\) be the homogeneous ideal of a projective variety \(V\), defined over an arbitrary algebraically closed field. The symbolic power \(I^{(m)}\) contains forms vanishing to a certain order, at general points of components of \(V\). The knowledge of relations between symbolic powers \(I^{(m)}\) and usual powers \(I^r\), especially when \(V\) is a finite set of points, would be of valuable help in the study of several interpolation problems. Unfortunately, these relations are widely unknown, even for low values of \(r,m\). For instance, in the case where \(I\) represents some set of points in \(\mathbb P^2\), the question whether \(I^{(3)}\subset I^2\) or not, is open.
To study pairs \((m,r)\) for which \(I^{(m)}\subset I^r\), the authors introduce the resurgence index \(\rho\) of \(I\) as: \[ \rho(I)=\sup\{m/r: I^{(m)}\not\subset I^r\}, \] which always exists, when \(I\) is radical. Then, the authors study relations between the resurgence and other invariants of the ideal \(I\). When \(V\) is zero–dimensional, the authors prove a bound for \(\rho(I)\), in terms of the minimal degree of forms in \(I\) and a Seshadri constant of the ideal. Several applications of the bound are listed. In particular, the authors show that \(I^{(3)}\subset I^2\), when \(V\) is a general set of points in \(\mathbb P^2\).