  We introduce certain canonical blow-ups $\mathcal T_{s,p,n}$, as well as
their distinct submanifolds $\mathcal M_{s,p,n}$, of Grassmann manifolds
$G(p,n)$ by partitioning the Pl\"ucker coordinates with respect to a parameter
$s$. Various geometric aspects of $\mathcal T_{s,p,n}$ and $\mathcal M_{s,p,n}$
are studied, for instance, the smoothness, the holomorphic symmetries, the
(semi-)positivity of the anti-canonical bundles, the existence of
K\"ahler-Einstein metrics, the functoriality, etc. In particular, we introduce
the notion of homeward compactification, of which $\mathcal T_{s,p,n}$ are
examples, as a generalization of the wonderful compactification. Lastly, a
generalization of $\mathcal T_{s,p,n}$ according to vector-valued parameters
$\overline s$ is given, and open questions are raised.
