Blow-up of a generalized flag variety

Abstract: Let $G$ be a connected simply connected semisimple complex algebraic group and $P\, \subset\, G$ a parabolic subgroup. We give a necessary and sufficient condition for a line bundle, on the blow-up of the generalized flag variety $G/P$ along a smooth Schubert variety, to be ample (respectively, nef). Furthermore, it is shown that every such nef line bundle is globally generated. As a consequence, we describe when such a blow-up is (weak) Fano.