  We study the geometry of equivariant, proper maps from homogeneous bundles
$G\times_P V$ over flag varieties $G/P$ to representations of $G$, called
collapsing maps. Kempf showed that, provided the bundle is completely
reducible, the image $G\cdot V$ of a collapsing map has rational singularities
in characteristic zero. We extend this result to positive characteristic and
show that for the analogous bundles the saturation $G\cdot V$ is strongly
$F$-regular if its coordinate ring has a good filtration. We further show that
in this case the images of collapsing maps of homogeneous bundles restricted to
Schubert varieties are $F$-rational in positive characteristic, and have
rational singularities in characteristic zero. We provide results on the
singularities and defining equations of saturations $G\cdot X$ for $P$-stable
closed subvarieties $X\subset V$. We give criteria for the existence of good
filtrations for the coordinate ring of $G\cdot X$. Our results give a uniform,
characteristic-free approach for the study of the geometry of a number of
important varieties: multicones over Schubert varieties, determinantal
varieties in the space of matrices, symmetric matrices, skew-symmetric
matrices, and certain matrix Schubert varieties therein, representation
varieties of radical square zero algebras (e.g. varieties of complexes),
subspace varieties, higher rank varieties, etc.
