2023/05/19 by Indranil Biswas, Biswas, Indranil, S. Senthamarai Kannan +3
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2305.11404
openalex publication_date 2023/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a semi-simple simply connected algebraic group over the field ℂ of complex numbers. Let T be a maximal torus of G, and let W be the Weyl group of G with respect to T. Let Z(w, \underlinei) be the Bott-Samelson-Demazure-Hansen variety corresponding to a tuple \underlinei associated to a reduced expression of an element w ∈ W. We prove that for the tuple \underlinei associated to any reduced expression of a minuscule Weyl group element w, the anti-canonical line bundle on Z(w, \underlinei) is globally generated. As consequence, we prove that Z(w, \underlinei) is weak Fano. Assume that G is a simple algebraic group whose type is different from A2. Let S = \α1, ⋯, αn\ be the set of simple roots. Let w be such that support of w is equal to S. We prove that Z(w, \underlinei) is Fano for the tuple \underlinei associated to any reduced expression of w if and only if w is a Coxeter element and w-1(∑t=1nαt) ∈ -S.