Blow-up of a generalized flag variety

TL;DR

This paper characterizes the ampleness and nefness of line bundles on the blow-up of generalized flag varieties along smooth Schubert varieties, providing criteria for when these blow-ups are Fano or weak Fano.

Contribution

It offers necessary and sufficient conditions for line bundles to be ample or nef on these blow-ups, and shows all nef line bundles are globally generated, advancing understanding of their geometric properties.

Findings

Criteria for ampleness and nefness of line bundles

All nef line bundles are globally generated

Conditions for the blow-up to be (weak) Fano

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.