Spaces of holomorphic maps between complex projective spaces of degree one

For an integer d⩾0, let Hol d ( C P k , C P n ) denote the space consisting of all holomorphic maps f : C P k → C P n of degree d. We study the homogeneous space structure of Hol d ( C P k , C P n ) for the case d=1. In particular we explicitly determine its homotopy type.