  We consider Monge-Amp\`ere equations with right hand side $f$ that degenerate
to $\infty$ near the boundary of a convex domain $\Omega$, which are of the
type $$\mathrm{det}\;D^2 u=f\quad\mathrm{in}\;\Omega,\quad\quad f\sim
d^{-\alpha}_{\partial\Omega}\quad\mathrm{near}\;\partial\Omega,$$ where
$d_{\partial\Omega}$ represents the distance to $\partial \Omega$ and $-\alpha$
is a negative power with $\alpha\in(0,2)$. We study the boundary regularity of
the solutions and establish a localization theorem for boundary sections.
