  We consider the inhomogeneous nonlinear Schr\"odinger equation (INLS) in
$\mathbb{R}^N$, $N \geq 1$, $$i \partial_t u + \Delta u + |x|^{-b} |u|^{p-1}u =
0,$$ with finite-variance initial data $u_0 \in H^1(\mathbb{R}^N)$. We extend
the dichotomy between scattering and blow-up for solutions above the
mass-energy threshold (and with arbitrarily large energy). We also show other
two blow-up criteria, wich are valid in any mass-supercritical setting, given
there is local well-posedness.
