2025/09/03 by I. Grojnowski, Grojnowski, I., N. I. Shepherd‐Barron +1
Mathematics · #14M15 #20G07 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.2509.03504
openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix a flat and projective morphism X→Σ of schemes. We show, first, that any set of ℙ1-fibrations on X defines a set of simple roots, a set of simple coroots and a Cartan matrix C. Second, X is an étale F-bundle over some projective Σ-scheme, where F is the flag variety of the adjoint Chevalley group over the integers defined by C. In particular, if the simple roots generate the Néron--Severi group of X relative to Σ and X is cohomologically flat in degree zero over Σ then X is a form of F. When X is a smooth Fano variety over the complex numbers all of whose extremal rays are accounted for by these fibrations this is due to Occhetta, Solá-Conde, Watanabe and Wiśniewski. Third, we recover, in a uniform way, the isomorphism and isogeny theorems of Chevalley and Demazure: over any base a pinned reductive group is determined by its pinned root datum, and a p-morphism of pinned root data determines a unique homomorphism of the corresponding groups.