2022/12/20 by Saurav Bhaumik, Bhaumik, Saurav, Pinakinath Saha +1
Mathematics · Physics and Astronomy · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2212.10366
Let G be a semi-simple simply connected algebraic group over an algebraically closed field k of arbitrary characteristic. Let B be a Borel subgroup of G containing a maximal torus T of G. Let W be the Weyl group of G with respect to T. For an arbitrary sequence w=(s_i1,s_i2,…, s_ir) of simple reflections in W, let Zw be the Bott-Samelson-Demazure-Hansen variety (BSDH-variety for short) corresponding to w. Let \widetildeZw:=G×BZw denote the fibre bundle over G/B with the fibre over B/B is Zw. In this article, we give necessary and sufficient conditions for the varieties Zw and \widetildeZw to be Fano (weak-Fano). We show that a line bundle on Zw is globally generated if and only if it is nef. We show that Picard group Pic(\widetildeZw) is free abelian and we construct a O(1)-basis. We characterize the nef, globally generated, ample and very ample line bundles on \widetildeZw in terms of the O(1)-basis.