  We prove the connectedness of the moduli space of maps (of fixed genus and
homology class) to the homogeneous space G/P by degeneration via the maximal
torus action. In the genus 0 case, the irreducibility of the moduli of maps is
a direct consequence of connectedness. An analysis of a related
Bialynicki-Birula stratification of the map space yields a rationality result:
the (coarse) moduli space of genus 0 maps to G/P is a rational variety. The
rationality argument depends essentially upon rationality results for quotients
of SL2 representations proven by Katsylo and Bogomolov.
