Blow-up of a generalized flag variety

TL;DR

The paper addresses when the blow-up ${\rm Bl}_{Z}X$ of a generalized flag variety $X=G/P$ along a smooth Schubert subvariety $Z$ is Fano or (weak) Fano. It provides a complete description of the nef/ample cones: they are generated by pullbacks ${\rm Bl}_{Z}^{*}D_{\alpha}$ and the exceptional class $H-E_{Z}$, with a dual Mori cone described by modified Schubert curves; ampleness, nefness, and global generation criteria follow from these cones. The anticanonical bundle is analyzed for bigness and Fano-type properties, yielding explicit conditions and special-case simplifications, including Grassmannians and cominuscule cases. These results generalize known Grassmannian blow-up phenomena via root-system data and offer practical, computable criteria for positivity and global generation on blow-ups.

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.

Theorems & Definitions (35)

• Lemma 3.1: cf. Hartshorne
• Corollary 3.2
• Theorem 3.3
• proof
• Corollary 3.6
• proof
• Lemma 3.7
• proof
• Theorem 3.8
• proof