Summary: We study the initial-value problem for the nonlinear Schrödinger equation \[ i \partial _t u+\Delta u=\lambda | u | ^p,\quad (t,x)\in [0,T)\times \mathbb {R}^n, \] where \(1<p\) and \(\lambda \in \mathbb {C}\setminus \{0\}\). The local well-posedness is well known in \(L^2\) if \(1<p<1+4/n\). In this paper, we study the global behavior of the solutions, and we prove a small-data blow-up result of an \(L^2\)-solution when \(1<p\leq 1+2/n\).