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 actually globally generated. As a consequence, we are able to describe when such a blow-up is (weak) Fano.