  In this paper we describe the structure of the space of parabolic reductions,
and their compactifications, of principal $G$-bundles over a smooth projective
curve over an algebraically closed field of arbitrary characteristic. We first
prove estimates for the dimensions of moduli spaces of stable maps to the
twisted flag varieties $E/P$ and Hilbert schemes of closed subschemes of $E/P$
with same Hilbert polynomials as that of a $P$-reductions of $E$. This
generalizes the earlier results of Mihnea Popa and Mike Roth to connected
reductive groups and the results of Y. I. Holla and M. S. Narasimhan to the
case of non-minimal sections. We then prove irreducibility and generic
smoothness of the space of reductions for large numerical constraints, using
the above result and the methods of G. Harder. We also study these space in
more detail for generically stable $G$-bundles. As a consequence we can
generalize the lower bound results of H. Lange to $G$.

Algebraic Geometry